#Class 12 Maths Chapter Integrals

[vavaclasses]

Integration by Parts – Class 12 Mathematics Notes

Introduction:

Integration by Parts is a crucial technique in integral calculus, especially useful when dealing with the integration of the product of two functions. It is derived from the product rule of differentiation and helps solve complex integrals that cannot be integrated directly. Understanding this method thoroughly will aid in solving various problems in CBSE Class 12 board exams and competitive exams like JEE Main.

Formula for Integration by Parts:

If u = f(x) and v = g(x), then:

∫ u·v dx = u ∫v dx - ��� (du/dx · ∫v dx) dx

Or simply,

∫ u·v dx = uv - ∫ v·(du/dx) dx

Choosing u and v – ILATE Rule:

To select which function to differentiate and which to integrate, use the ILATE rule:

I: Inverse Trigonometric functions L: Logarithmic functions A: Algebraic functions T: Trigonometric functions E: Exponential functions

Solved Examples of Integration by Parts:

Evaluate ∫ x · e^x dx

Let u = x (Algebraic), dv = e^x dx Then, du = dx, and v = ∫ e^x dx = e^x Apply the formula: ∫ x·e^x dx = x·e^x - ∫ e^x dx = x·e^x - e^x + C Answer: ∫ x·e^x dx = e^x(x - 1) + C

Evaluate ∫ ln x dx

Let u = ln x, dv = dx Then, du = (1/x) dx, v = ∫ dx = x Apply the formula: ∫ ln x dx = x·ln x - ∫ x·(1/x) dx = x·ln x - ∫ 1 dx = x·ln x - x + C Answer: ∫ ln x dx = x(ln x - 1) + C

Evaluate ∫ x · sin x dx

Let u = x, dv = sin x dx Then, du = dx, v = ∫ sin x dx = -cos x Apply the formula: ∫ x·sin x dx = -x·cos x + ∫ cos x dx = -x·cos x + sin x + C Answer: ∫ x·sin x dx = -x·cos x + sin x + C

Special Cases and Tips:

Some integrals may require repeated application of the formula. For example: ∫ x^2 e^x dx

Practice Questions

1. ∫ x · cos x dx 2. ∫ x · ln x dx 3. ∫ x^2 · e^x dx 4. ∫ arctan x dx 5. ∫ ln x dx

Conclusion:

Integration by Parts is a powerful technique in calculus, especially when dealing with products of functions. Mastery of the ILATE rule and regular practice of varied problems ensures confidence and accuracy in the exams.

[edupunkn00b]

Meus ex Machina, Chapter 12: Working Out

Edited public domain image of two hands reaching for each other, lit in deep blue and neon green.

Prev - Working Out - Next - Masterpost - [ AO3 ]

Logan is determined not to slow down the team again. It wouldn't be the first time he'd pushed his body to the limit.

Logan slipped through the locker room door just before it clanked shut, closing in three dozen or so students in the musky, humid air. "Watch it!" he warned a knot of augmented juniors as they chased past, jostling his safety glasses. His cry went unnoticed over their excited laughter.

He sidled close to the only other full Traditional he could spot, a tall, muscular sophomore. The boy may not have been born with Powers, but he’d certainly clutched a winning genetics lottery ticket on his way into the world.

“Hey, Lo, check it out!” Andrei cheered, lifting his shirt to reveal a translucent implant above his right hip. “Look what my parents got me for my birthday!” Nodding, Logan watched the lights flicker, micro-adjusting his hormone and electrolyte levels. “No more charley horses for me, huh?”

“Indeed,” he murmured. No more leg cramps, no more muscle fatigue. And with a daily boost of somatropin, he’d grow at least another foot and a half before he graduated. “Congratulations.” Logan forced a smile, adjusting his now-fogged eyeglasses. “You will certainly—”

“Alright boys, listen up!” The wrestling coach stepped up onto a nearby bench and whistled. Two freshmen with new hearing augments winced and covered their ears. He waited for the din to lessen and tapped his wrist, activating the ceiling's vid projector. Thanks to Title IV of the Powerds Integration Act, blah blah blah,” he rolled his eyes and two of the older Powerds glared at him. “We’re down to one boy’s team this year with space for five Traditionals.”

Five? Logan's stomach dropped to his feet. There had been fifteen Traditionals in last year’s team plus four alternates. He scanned the room as the coach called out the new team’s roster. Jake and Kirthi high fived behind the coach, reaching nearly his height now with the summer’s growth injections taking effect. Ben stood in the corner, dark shades covering new eyes.

Logan's mouth went dry when he realized he was the last of the Traditionals team who hadn’t yet been augmented. Tim wasn’t there, of course, though Logan saw him from time to time between classes, slouched in his motorized chair, waiting for the school’s singular elevator.

“And… Sanders,” the coach finished. Thirty four sets of eyes whipped over to him and Logan tried to stand a little taller. And tried to ignore the other student’s murmurs.

“It’ll be a good year, Coach,” he said.

Coach Roberts nodded, eyes fixed at a spot just above his head. “Suit up, boys. I want you all out and sparring in pairs in five.” 

Logan kept his eyes trained down and focused on his own preparations as the locker room exploded in activity. Laughter and slamming doors rattled his teeth, the usual scramble to shed street clothes and don stretchy singlets—or, in the case of several boys, trade in for the next size up—no longer lending him its usual energizing buzz.

The day wasn't finished with him yet. With an odd number of Traditionals on the team, it was a simple matter of the math not mathing to require a Traditional to that be paired with a Powered.

Coach Roberts patted Logan’s shoulder as he stepped up to his side of the mat. “Look, Sanders, do your best out there. No-one expects much more than that from you.”

Logan looked up into Coach Robert’s camera-ringed irises. “Y—yes, Coach,” he nodded and turned to face his opponent. The coach had at least tried to give him a chance, matching Logan with a Powered in the same weight class.

But how much did weight class really matter when your opponent’s muscles powered a skeleton stronger than titanium? When your opponent could sense your thoughts, predicting your every move?

He managed to stay on his feet for the first round. Until, at least, his sparring partner grew bored and stopped tamping down on his other abilities. In seconds, Logan was down, face jammed into the mat. He struggled but his opponent countered each twist, each desperate attempt to break free. Finally, Logan tapped out.

The Powered—a new student Logan hadn’t even learned his name—loomed over him. “Good job.” He tonelessly repeated the team’s end of match phrase and looked around for his next partner.

Logan pushed up to his feet and wiped the blood from his mouth with the back of his hand and shook his head. “Best two out of three.”

~

“Time?” Logan panted, controller shaking in his grip as he brought the mech's hands together in the ready signal. He’d boosted the suit’s stabilizers, so from the outside, at least, his durasteel arms moved smoothly. But Logan knew. 

“219.8 seconds. Better,” V smiled encouragingly but the flatness in his tone and the twitch in his eyebrow said it all. Logan still took entirely too long to properly suit up. 

He deflated, leaned to one side in the mech’s cavity. Logan pulled in a shaking breath, fighting to slow his panting, and nodded. “Indeed. Better.” Better but not enough.

"Get your bearings and we’ll go again,” V suggested, still smiling as he passed him a towel. It was only then Logan noticed he was drenched, sweat dripping from his hair and soaking through his shirt.

Logan smiled his thanks—tried to, at least, and V nodded again, a bottle of electrolyte ready. His smile grew, just a bit. He hadn't yet exhausted V's patience. Logan would keep going for as long as he was willing to help him.

V looked up when Patton approached from the kitchen, hugging a large crate to his burly chest. The scents of strawberries and fresh bread and chocolate chip cookies intermingled in the air as he stopped. He shifted the box to one arm to wave at them. “I’m headed, ah…” He waggled his head vaguely toward the hall and Logan’s brow furrowed before understanding struck.

“Is he…” V frowned and swiped across his tablet, searching. Trying not to be obvious, Logan peeked over his shoulder and caught a glimpse of The Muse’s hallway on the screen.

“No, he—nothing’s wrong. Just…” Patton shrugged and looked down at the crate. “After yesterday, I…”

Hesper’s cryptic words from last night filled Logan’s mind. “What makes you think I didn’t trigger your alarms so I could have a little company… just like Re used to?”

“Can I help?” Logan blurted out before looking down at his useless half a hand, his broken body. His face burned, stomach knotted in shame at his utter uselessness.

Patton’s smile was gentle. “Um, that’s okay, Kiddo. I got this.” He didn’t bother to ask how precisely Logan thought he could help in a room he couldn’t reach in his mech. In a room he couldn’t even enter without assistance. Still holding the giant crate of food with one arm, he gently squeezed Logan’s—not the mech’s—shoulder. “But thanks. I’ll let you both get back to your drills.”

He gave them another little wave then headed down the hall. Listening to the elevator door open then swish closed, Logan finished his drink and passed V the empty bottle.

“Ready?” he asked, switching back to his timer.

“Yes.” Logan lowered the mech to a crouch and began to climb out. But Patton's arrival broke the dam on the flood of questions from last night. In his distraction, he nearly fell out of the cavity. “Well… wait.” He met V’s questioning glance. “What did Hesper mean… about The Prince as a little boy? Did he… Did Hesper know him back then?” V scowled down at his tablet but Logan pushed on. “Is that why he called The Muse, ‘Re?’”

Tapping at the screen, V started to speak, then his jaw clamped shut. He let out a slow breath before looking back at Logan. “It’s not really my story to tell. I…” He sighed again. 

For a moment, Logan feared V would try sending him to The Prince with his questions. “Yeah,” V said at last. “Yeah, there’s… there’s a history. You…” V winced and wouldn’t quite meet his eyes. “Maybe ask Silvertongue… um… later, though.” He nodded like it was obvious why he should wait. “You know?”

Was there more to Silvertongue’s outburst last night, too? “I’ll wait,” he promised. “Thank—thank you.” 

“Yeah.” V blew out a short breath and returned his attention to the tablet. “Ready for another go?”

Logan dropped the last foot or so, hitting the ground with a barely muffled groan. He climbed up into his chair, replicating his starting point. “I’m ready,” he said, looking up at the suit.

V stared at him for a long moment. “You know, maybe…” He looked away when Logan tried to meet his eyes, tugging at his sleeves.

“Maybe what?” Logan prompted, bracing himself for what passed for yet another bit of ‘constructive’ criticism.

“Maybe you just need to work on your strength a bit,” he said with a little shrug. “If you want, I could show you how to modify a couple of the machines in the fitness center to—”

“Show me.”

That was how Logan ended up strapped to the pull-up machine in the team’s fitness center, the bright automated trainer chirping out his reps into a sweat-proof earpiece. “And twenty. Good job! You’ve finished your workout—”

“Again!” Logan snapped, tugging at the bar as the platform lowered him to the floor.

“You have completed your recommended three sets of twenty, Logan,” the interface chirped back. Though its tone never changed, Logan couldn’t help but flinch at the sound of his own name. Constructed from heavily modded tech ‘borrowed’ from Abracadabra, the interface sounded just like the old assignment program back at the distribution center. 

“I don’t care. Again!” He batted at the controls with the end of his left stump but the panel went dark.

“I am unable to comply with your request.” The artificial voice switched over to the main speakers when it ended his session, like it wanted a witness. “Please try again tomorrow.”

“You’re better off arguing with the refridgerator, Tin Man,” the Prince chuckled from the corner.

“How long have you been watching me?” Logan snapped, face warm. While the automated trainer didn’t record his strained grunts during his workout, the Prince was the absolute last person he’d want to hear him struggling to do a simple pull-up.

The Prince lifted a barbell packed with 300 pounds on either side. The weight floated up to his chin, before he raised it up above his head and back down to the floor in one perfect, flowing motion.

The weights didn’t even clank when they touched the mat.

“Long enough to see you max out your reps,” he said, barely pausing for his own weight routine. “Hit the showers or something.”

“I’m not done,” Logan muttered, pushing himself off the platform. Fatigued muscles trembling under the strain, he dropped faster than he’d planned, landing with a pained grunt as his hip grazed the base of the pull up machine. 

“Don’t tell me it’s leg day for you.”

“Oh, ha, ha,” Logan muttered, hiding his face until he could control the shameful quiver in his chin. “With wit like that, it’s little wonder you’re the brains of the team. Oh, no, wait—You’re not.”

The Prince shrugged and spared him any further conversation, instead loading on another 100 pound plate to each side of his dumbbell. Shaking his head, Logan pulled himself into his chair and moved to the far corner of the fitness room, keeping as many weight machines between him and The Prince as he could.

When he’d found a suitable place, he lowered himself to the floor again, taking it slowly this time. Laying flat on his back with his chair in front of him, he wedged his thighs between the front and back wheels, bracing his lower body for crunches.

It worked well. For a while, at least. To remain steady, Logan had to tighten his abductors with each crunch, pushing against the wheels to keep them—and himself—from rocking with the movement.

He pushed on, the chair straining against its emergency brakes with each rep. Finally, the brake snapped free and the chair flipped back.

“Fuck.” Logan pounded his fist and the back of his head against the mat and lay back, eyes closed, as he gathered the strength to push the chair upright. With any luck, the Prince had already finished his strongman routine and left the gym.

Logan was not a lucky man.

“Need a hand?”

Logan glared up at him from the floor.

“Oh, shit, yeah…” He shuffled closer, ears tinged pink as he avoided looking at Logan’s stump. “I—I didn’t mean that one. I swear.” He gestured at the overturned wheelchair. “Can I help?”

“I’ve got it,” Logan muttered, pushing up and wiggling closer to the chair. He rolled onto his belly then wormed his way into the gap between the back of the chair and the floor. Once he’d gotten most of his body underneath it, he wedged his right hand under his chest and pushed up sharply.

It took two more tries but finally the chair tilted forward and Logan twisted with it, grabbing the strap criss-crossing the back of the chair to steady it before it tipped completely forward.

“Damn,” the Prince murmured.

“You’re still here?” Logan leaned against the seat back, panting, and in no way ready to either resume his crunches or climb into the chair.

The Prince shifted his weight from side to side, one foot pointed toward the door, the other toward him. Finally, he dropped to the floor and sat about two feet away. Jerking his chin toward the chair, he shrugged. “What were you even doing to knock over that beast?”

Logan eyed The Prince and considered ignoring his question. If he really was that nosy, he could just as easily pull up the security footage and that might be worse than simply hearing about it. Sighing, Logan let his eyes fall closed. “I was using it to brace my… legs for abdominal crunches. The other machine's not built for me.”

The Prince made a little choked sound in the back of his throat. Not a laugh, but…

Eyes still shut, Logan wiggled back into position and prepared himself for another set. If he could increase his core strength, maybe the one-armed pull-ups would get easier faster.

Logan’s eyes snapped open when he felt movement next to him. The Prince had moved closer, knelt on the floor only about a half-foot away, hands laying open on his lap. His expression was unreadable.

“Lay your legs against mine. I’ll brace you,” he said, patting the tops of his thighs and glancing down at where Logan’s legs ended. To the Prince’s credit, the usual mix of pity and disgust flashing over his features didn’t last long.

The Prince waited, holding his gaze until, finally, Logan nodded. He moved a little closer, but let Logan be the one to move close enough to touch and set his legs in place. “May I?” he asked, holding his hands over the tops of Logan’s thighs.

“Go—” Logan cleared his throat, the vulnerability of his position seizing the muscles in his neck. If he'd wanted to, The Prince could pick him up and fling him against the wall and not even break a sweat. “Go ahead,” he finally croaked out. With flat, open hands, The Prince pressed against his legs, holding him in place.

“I’ll let go the moment you tell me to,” the Prince said, then fell silent as Logan resumed his crunches.

He loathed to admit it, but focusing on only one set of muscles made the crunches almost enjoyable. Almost. “This helps,” Logan grunted. “Immensely.” He forced himself up for another two crunches. “Thank you.”

“Anytime,” The Prince replied, repeating his own words back at him from last night's skirmish with Hesper.

Logan pushed out the last of the set and lay back, counting down a thirty second rest period. He lifted up again and silently worked through another set. He slowed as he neared the end, muscles screaming in protest. “Why…" The temptation to ask The Prince about his history with Hesper—and about Re—was great. He shifted to the next question bubbling in his mind instead. "What are you helping me, anyway? I thought… I thought you hated me.”

“I don’t—” The twitch in The Prince’s jaw told Logan he very much did. “We’re a team,” he shrugged at last. “If you’re stronger, we’re all stronger. C’mon, Iron Man, that all you got?" Did Logan only imagine a smile behind his eyes? "Gimme three more.”

[fitzrove]

Passing this on :p

Fic authors self rec! When you get this, reply with your favorite five fics that you've written, then pass on to at least five other writers. Let’s spread the self-love! 💞

Ahhhhh thank you so much!!!!!! This is so nice of you to send >:] tysm <3

I'm going to recommend both Elisabeth and Endeavour fics because I've spent a lot of time writing for both!! So let's get to it, in no order of preference:

Feel The Night - Endeavour (TV) - rated T, Morse x Jakes, vampire AU, Victorian setting, mystery/romance, ballroom dancing, neck biting, strangers to lovers, 5 chapters / 23.5k

This is secretly a TdV / Endeavour fusion (from before my full-blown euromusical era hehehehehe). I think it's some of the best work on setting and mood that I've done - like I did a bunch of research and actually put that to the page ajdjjjg. Really proud of it even 3 years down the line!!

Flights of Angels - Elisabeth - rated M (references sex), Rudolf x Tod, post-canon, angst with a happy ending, character growth, historical references and commentary, 12 chapters / 20k

My todolf divorce fanfic xD Exploring the deep implications of the common trope of human-to-Todesengel; Tod backstory and worldbuilding, esp referencing the human-metaphor dichotomy inherent in Máté's portrayal; character growth and character development for Rudolf. Yeah I wax poetic but this is actually mostly an exploration of how Máté!Tod and Lukas!Rudolf (because they are very specific portrayals!!!!) could make a long-term relationship work. There are Implications to it. Also, Mizzi Kaspar appears!!!

Vögelein - Elisabeth - rated M (but one chapter might as well be E), Rudolf x Tod, lesbian todolf, 5+1, historical commentary and references, 6 chapters / 7.2k

Lesbian todolf!!!!!!!!!!!!!!!!!! It's literally just "what if Rudolf was born a girl" (and Stéphanie a boy) and goes from there - ie. we end up in the Belgian court which comes with extremely unfortunate implications in the late 19th century. Also Tod is a woman (hehe lyrics reference) because I don't stand for heterosexuality sorry AHSHFHGH. Rudolf also keeps all his (= her) unsavoury historical womanizing because it's so integral to the character, in my opinion. I'm really proud of this >:]

Soft Like Summer Rain - Endeavour - rated M (but one chapter might as well be E), Morse x Jakes / Jakes x Hope / Morse x Jakes x Hope, mutual pining, 21 chapters / 51.6k

My longest fic to date!! I wrote it when I was 18 fun fact: largely in class too xD so some parts I can't bear to reread because I'm afraid I'll find lots of melodrama ahdjg, but it's one of a kind and I'm so proud of what it represents!! Plot: what if Morse visited Jakes over the summer after S3 and things kinda went from there. I can't believe Jakes asking him to come to the US with him actually became canon in S9, 4 years after this fic was published,, half-convinced Russ Lewis reads my fanfic...

Midnight Man - Elisabeth - rated E, Rudolf x Tod, modern AU, yeah um it's a one-shot that's rated E and only 2k words long you can do the math about what happens in it...

My first todolf fanfic xD Lawyer Rudolf. There's an entire unpublished modern AU universe based on this but it's more fun to think about than it necessarily is to write out, especially since the fun parts are just Tod forcibly inserting himself into Rudolf's life as common-law boyfriend/roommate (depression metaphor!!1) and getting up to shenanigans. I don't really have interesting things to say re: it being a modern AU, which makes it harder to write. I've worked on a ballet AU behind the scenes for like 2 years adjhfjjg which is a modern AU that takes the metaphors a lot further... but I haven't managed to write it because I'm not entirely sure how I'll approach it, especially the Tod characterisation in that. But I've done so much research that hopefully I will write something in that universe eventually...

Thank you again for sending this, it was so much fun to look back on what I've written!! >:]

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[katbisgaard]

Ashes Chapter 1 Part 12

“You’re making friends.”

Garrett nodded, even though it didn’t feel true. He was making connections, allies. Friends were something else.

“Good,” Lionel said. He was always sitting with a notebook in his lap, but he rarely wrote anything down, although it always seemed to be full of writing. He must write his notes when the session was over, but he still held a pen in his hand the entire time. At least the way he fiddled with it gave Garrett something to focus on, when he didn’t want to look at Lionel.

“You haven’t had anymore panic attacks?”

“No.” He’d only had the one, and he still didn’t know why. Even Lionel hadn’t been able to coax that truth out of him. Although, if he was being honest, it probably had more to do with the trauma of remembering his manifestation, than it had to do with his former home and his former family. In the end, Lionel had been forced to conclude that it had been a one time thing, and Garrett didn’t suffer from anxiety. The conversations had continued regardless, but this could be the last. Garrett had been given the choice. It was no longer obligatory, but it was up to him to say stop. He wasn’t sure. He had wanted to hate it, just on principle, but it hadn’t been all bad. It had been nice, even, to have someone to talk to. Someone who didn’t get to tell anyone else.

“What was your favorite class this week?”

They settled into the usual line of questioning. It had been hard at first, analyzing his own week in likes and dislikes, but it had gotten easier with time. He didn’t know exactly what the point was, except to identify the good and the bad. The things that he enjoyed and the things he had to endure. Sometimes, the dislikes would make him realize, or make Lionel point out, that it was possible to make changes, to not endure things you didn’t like, and sometimes it wasn’t. The important part was to identify the things you could change, and decide to make the effort.

Meanwhile, he used the good things to illustrate that sometimes good things didn’t last. It wasn’t like Garrett wasn’t acutely aware of that, but he allowed Lionel to put it into perspective for him. And, the lesson he preferred: Sometimes, you just had to congratulate yourself on your own success.

It was nice to be acknowledged, and to know that he didn’t need other people’s approval to succeed. It was enough that it just made him feel good about himself.

“Intro to magic,” he answered, to the first one. It had been the same answer nearly every time. Everything else was high school classes, and even getting an A in math couldn’t compete with learning about magic. “We learned about wild magic today.”

“And you enjoyed that?”

Garrett thought back to that red thing he had coughed up. He thought about the Healer’s words: It just wants your fear. He closed his eyes. “I was nice to… know.”

Lionel nodded. Garrett thought he would be asked to elaborate, but they moved on.

The questions flowed. Least favorite class? Where had he felt success? Failure? Best thing he had done for himself? Worst thing? Then, “What was your favorite interaction this week?”

“What?” Garrett’s eyes flickered from the pen to Lionel. “You haven’t asked me that before.”

Lionel smiled. “You weren’t ready for that question.”

He was expecting something about his friends. Garrett scrambled to remember a single conversation he’d had with any of them. They had taken him in, but they hadn’t really accepted him.

“Grace,” he said, because it was the easy answer. Because she was the one who had reached out. “She’s been… bringing me into the world more.” The group was together a lot. They ate their meals together, they studied together, they went to coffee shops. It was exhausting, but it was necessary. He wanted to integrate. He needed to. He had never been this much of a social outcast before, and it was time to stop hiding.

“I’m glad.” Lionel closed the notebook and put down the pen. “Now, is there anything else on your mind?”

This one always felt like a trick question.

“No,” he answered, and felt like he was failing a test.

Lionel had expected it, since it was the answer he gave most times.

“Okay, then there’s just one thing left: Will I see you next week?”

Garrett opened his mouth to answer. Closed it. He’d thought about it a lot. He should have been ready to commit to either option. “I can quit anytime?”

“Anytime,” Lionel confirmed.

Garrett nodded. “Okay. Same time next week?”

“See you then.”

Lionel looked a little too pleased about it, but Garrett let it slide. Having someone like him choose therapy must be how someone like Lionel measured his own success.

[mpboardsolutions-blog]

MP Board Class 12th Maths Book Solutions in English Medium

MP Board Class 12th Maths Chapter 1 Relations and Functions

Chapter 1 Relations and Functions Ex 1.1

Chapter 1 Relations and Functions Ex 1.2

Chapter 1 Relations and Functions Ex 1.3

Chapter 1 Relations and Functions Ex 1.4

Chapter 1 Relations and Functions Miscellaneous Exercise

MP Board Class 12th Maths Chapter 2 Inverse Trigonometric Functions

Chapter 2 Inverse Trigonometric Functions Ex 2.1

Chapter 2 Inverse Trigonometric Functions Ex 2.2

Chapter 2 Inverse Trigonometric Functions Miscellaneous Exercise

MP Board Class 12th Maths Chapter 3 Matrices

Chapter 3 Matrices Ex 3.1

Chapter 3 Matrices Ex 3.2

Chapter 3 Matrices Ex 3.3

Chapter 3 Matrices Ex 3.4

Chapter 3 Matrices Miscellaneous Exercise

MP Board Class 12th Maths Chapter 4 Determinants

Chapter 4 Determinants Ex 4.1

Chapter 4 Determinants Ex 4.2

Chapter 4 Determinants Ex 4.3

Chapter 4 Determinants Ex 4.4

Chapter 4 Determinants Ex 4.5

Chapter 4 Determinants Ex 4.6

Chapter 4 Determinants Miscellaneous Exercise

MP Board Class 12th Maths Chapter 5 Continuity and Differentiability

Chapter 5 Continuity and Differentiability Ex 5.1

Chapter 5 Continuity and Differentiability Ex 5.2

Chapter 5 Continuity and Differentiability Ex 5.3

Chapter 5 Continuity and Differentiability Ex 5.4

Chapter 5 Continuity and Differentiability Ex 5.5

Chapter 5 Continuity and Differentiability Ex 5.6

Chapter 5 Continuity and Differentiability Ex 5.7

Chapter 5 Continuity and Differentiability Ex 5.8

Chapter 5 Continuity and Differentiability Miscellaneous Exercise

MP Board Class 12th Maths Chapter 6 Application of Derivatives

Chapter 6 Application of Derivatives Ex 6.1

Chapter 6 Application of Derivatives Ex 6.2

Chapter 6 Application of Derivatives Ex 6.3

Chapter 6 Application of Derivatives Ex 6.4

Chapter 6 Application of Derivatives Ex 6.5

Chapter 6 Application of Derivatives Miscellaneous Exercise

MP Board Class 12th Maths Chapter 7 Integrals

Chapter 7 Integrals Ex 7.1

Chapter 7 Integrals Ex 7.2

Chapter 7 Integrals Ex 7.3

Chapter 7 Integrals Ex 7.4

Chapter 7 Integrals Ex 7.5

Chapter 7 Integrals Ex 7.6

Chapter 7 Integrals Ex 7.7

Chapter 7 Integrals Ex 7.8

Chapter 7 Integrals Ex 7.9

Chapter 7 Integrals Ex 7.10

Chapter 7 Integrals Ex 7.11

Chapter 7 Integrals Miscellaneous Exercise

MP Board Class 12th Maths Chapter 8 Application of Integrals

Chapter 8 Application of Integrals Ex 8.1

Chapter 8 Application of Integrals Ex 8.2

Chapter 8 Application of Integrals Miscellaneous Exercise

MP Board Class 12th Maths Chapter 9 Differential Equations

Chapter 9 Differential Equations Ex 9.1

Chapter 9 Differential Equations Ex 9.2

Chapter 9 Differential Equations Ex 9.3

Chapter 9 Differential Equations Ex 9.4

Chapter 9 Differential Equations Ex 9.5

Chapter 9 Differential Equations Ex 9.6

Chapter 9 Differential Equations Miscellaneous Exercise

MP Board Class 12th Maths Chapter 10 Vector Algebra

Chapter 10 Vector Algebra Ex 10.1

Chapter 10 Vector Algebra Ex 10.2

Chapter 10 Vector Algebra Ex 10.3

Chapter 10 Vector Algebra Ex 10.4

Chapter 10 Vector Algebra Miscellaneous Exercise

MP Board Class 12th Maths Chapter 11 Three Dimensional Geometry

Chapter 11 Three Dimensional Geometry Ex 11.1

Chapter 11 Three Dimensional Geometry Ex 11.2

Chapter 11 Three Dimensional Geometry Ex 11.3

Chapter 11 Three Dimensional Geometry Miscellaneous Exercise

MP Board Class 12th Maths Chapter 12 Linear Programming

Chapter 12 Linear Programming Ex 12.1

Chapter 12 Linear Programming Ex 12.2

Chapter 12 Linear Programming Miscellaneous Exercise

MP Board Class 12th Maths Chapter 13 Probability

Chapter 13 Probability Ex 13.1

Chapter 13 Probability Ex 13.2

Chapter 13 Probability Ex 13.3

Chapter 13 Probability Ex 13.4

Chapter 13 Probability Ex 13.5

Chapter 13 Probability Miscellaneous Exercise

[ncertmcq]

Application of Integrals Class 12 Notes Maths Chapter 8

[meritbatch]

NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals

[integrals-calculus]

In this self-study course, you will learn integration as inverse process of differentiation. Integration of a variety of functions by substitution, by partial fractions and by parts, Evaluation of simple integrals, properties of definite Integrals and their derivations including answers to all NCERT exercise questions, examples, supplementary exercise questions and Sample Papers for NCERT Chapter 7 Integrals Class 12 Maths.

Integrals

Properties of Integrals

Class 12 Maths Solutions

[ashishkumarletslearn]

Integrals NCERT Chapter 7 Exercise 7.4 | Integration Class 12 - MathYug

Detailed explanation of questions related to NCERT Solutions for Exercise 7.4 (Q1 & Q9) of Integrals Class 12 Maths Chapter 7.  

Main Website: https://mathyug.com 

Mathyug is an online education platform focussed to strengthen your mathematical skills and allows you to deepen your knowledge by studying independently. 

[dennislabs]

How To Solve Class 12 Hard Chapters Easily

How To Solve Class 12 Hard Chapters Easily

Out of 13 sections in class 12 maths there are 3 hardest parts. They are complex and tedious. Simultaneously they are exceptionally scoring examples as well. Subsequently, this blog has been arranged that will allow 12th standard students like you to be aware, which is the hardest section in NCERT Solutions for Class 12 Maths? Three Toughest Chapter Of Class 12 Maths Applications of Integrals,…

[mathyug]

Detailed explanation of Direct & Simplification Method and questions based on NCERT Solutions of Chapter 7 Integrals Class 12 Maths IIT JEE Mains explained by expert teacher.

[aperrywilliams]

A Normal Conversation Ch07 (Spencer Reid x Maxine Brenner)

Previous Chapter / Ch07 / Next Chapter

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Chapter 7: The second date

Summary: Second date for Spencer and Max

Word Count: 2332

Rating: Teen And Up Audiences

--------------------

As he promised, Spencer was out of Max's school at 12:00 noon. Waiting in the same place as the last time. A few minutes later saw Max come out through the gate. Upon making eye contact they both smiled to each other. She walked over to where Spencer was.

"Hey… how are you?" said Max. “On time as ever Spencer" added.

"Hello. Yes, it's a habit on me... in case you hadn't noticed". Both grinned.

"It is a good habit I think. I admit it’s more hard for me..." Max said scratching her head.

"How have you been this morning? Are you ok?". She thought it was sweet of him asked her about it. She remembered their conversation from the night before over the phone.

“Yes, truly. I think I found myself just at a moment a bit distressing... but I'm fine now". Max said. “By the way... thanks. It helped me we could talk last night”.

"And it's what I intend us to do today" Spencer statement. “I have so many questions for you...". Max upped her eyebrows.

"Is it going to be some kind of FBI-style interrogation?" she joked. Spencer couldn't help but laugh.

"I hope not that kind of interrogation... shall we go?" he said quickly.

"Yeah, let’s go. We will walk there?”

"No, we’ll go in a cab. The place is a little further from here". After saying it, Spencer stopped a cab in front of them. He opened the door and invited her in first. Then he came in, closed the door, and the car moved when Spencer gave the driver instructions to get they were going.

Along the way Max told him the story of how she came art’s teacher at her school. Spencer thought it was reasonable she accepted the job in that circumstances.

"It was a good opportunity for you" he said.

“Yes, I needed to have my space and be independent and I think it was the best reason for me to make that decision. I had a good letter of recommendation too. That sure worked” said Max.

"But now you're not so comfy there..." said Spencer.

"I think I’m not" said Max, frowning. "And I think is because there is so much I could do with those children! But every time I propose something new I have someone at school who tells me that 'you can't', 'it's irrelevant', 'you have to worry about other things'… And so over and over again".

"Have you thought about what you would like to do, truly?" Spencer asked.

"So many times you would be surprised..." said Max. At that moment the cab stopped. They had reached their destination. Spencer paid and got off first, helping Max get off later.

They were in front of an Italian restaurant. Max looked at Spencer and couldn't help but joke.

"This time you’ll make sure you don't get hungry" They both laughed.

"Yeah, I wasn't going to take that risk again" said Spencer.

They walked towards the entrance and come into the place. It was not a big place, but it was very well decorated, very Italian style. Black and white photos of different people hung on the walls, possibly the owner's family of origin. There were checkered tablecloths on all the wooden tables. Chairs was wooden made too. A young woman approached Spencer and Max asking if they needed a table. Spencer nodded saying "for two, please". The young woman led them to one of the tables by a window, the one that faced the crowdy street. She handed them two menus and told them that the waiter would be taking their order soon.

They sat down and took the menus to see what they were going to eat. Max looked at the menu with an infinity of dishes on it and was not quite sure what to order.

"Any suggestions? So many options overwhelm me" Max asked. Spencer thought for a minute.

“If you ask me, I think the risotto at this place is fantastic. Although I traditionally choose pasta, I like it more”. At that point the waiter came to take their order. Max heeded Spencer's suggestion and chose a risotto with four cheeses sauce, while Spencer ordered his traditional pasta with putanesca sauce. To drink they asked for water and juice.

“I think I had already asked you in the cab, but now we could return to the subject. What would you like to work on?" Spencer asked.

"I've thought so many things" said Max, sighing. "From the most unlikely to very simple ideas". Spencer looked interested, waiting for her to continue talking. "For example, did you know in museums and theaters there are programs that encourage children's bond with arts as a way to integrate soft skills they cannot acquire at school?".

"I suppose it's reasonable to think of something like that given museums and theaters aren't going to survive long in this digital era otherwise" Spencer said.

"Sure, it may be a business plan for them, but despite it I think this kind of measures can help expand children's minds and connect them with things more sensitive than math’s... I would love to do something like that...". Saying it Max paused for a second. "Of course... no offense" finished saying. Spencer smiled at the last sentence.

"Are you saying it because I like math?... Ok, I think I can forgive you that" he said jokingly. Max smiled.

"I've also had crazy ideas like going to Los Angeles for artistic painting on the shore of the beach, smoking weed, like I did when I was younger...". Spencer stared at her trying to know if what she was saying the truth or teasing him. She noticed it and decided to challenge him. "Okay, Dr. Reid, Mr. Profiler... those I just said, is it true or a lie?". Spencer accepting the challenge scrutinized her eyes and body language. He watched her carefully and silently for a few minutes. Max couldn't help but blush feeling his eyes so meticulously look on her.

"It is… true!" Spencer sentenced, surprised himself by his finding. Max smiled sheepishly.

"Clearly there are things I did in my youth I'm not very proud of today ..."

"I am not going to judge you. I have no moral for it. I've also done things I'm not proud of…”said Spencer. At which he quickly regretted saying it. Max was about to question him when the waiter arrived with their order. "Our dishes!" Spencer said quickly "…Now you’ll tell me if my recommendation was good for you or not"

Max didn’t want to do her own questionnaire on him. She was hungry and preferred to focus on food. Apparently Max's plate had been a good recommendation, since he was enjoying it with great need. Spencer was also concentrated on his own plate. From moments to time he looked up to see how Max was enjoying her food.

"I owe you one. It was an excellent suggestion” said Max taking a sip of the juice that had been brought while they was eating.

"I’m happy you liked it" said Spencer pleased. After she wiped her lips with the napkin. Max started to speak again.

"Well then... FBI. And you travel a lot you told me. Do you like it?"

"My job? Too much. I love it. Travel is something you get used to… or at least you should get used to” said Spencer.

"And if you do profiles to catch murderers... isn't that like 'playing' at being them and thinking like them?"

"In fact, that's what we do". Spencer noticed Max frowning. "I know... it could be heard as very morbid thing... but the truth is... being difficult and unpleasant as it could be... sometimes... I think someone has to do it".

"I am not going to question it. I think it as a very dedicated job and it is good there are people like you who do it…”. Spencer looked at her knowing there was a ‘but’.

"But…?" Spencer asked.

"The thing is... how can you deal with it?... I mean, in your daily life". Spencer thought about it a few seconds before answering.

"I don't know... I guess trying not to think about it too much. At least I try… ” Spencer said.

"But you have those dreams..." said Max. ‘Touché…’ thought Spencer.

"Eh… yes. I think sometimes it just beats you... a little...". Spencer lowered his head and thoughtfully for a few moments. Max knew had struck a tough matter so she wanted to change the subject.

"Okay. I think we shouldn’t have to talk about work" She looked at him and gave him a smile. He smiled too. "What do you like to do when you are not traveling?"

"Wow... ok. I love to read, a lot. I have a small and interesting collection in my apartment”

"I can imagine it... and I bet it's bigger than you say, since I think it couldn't be any other way" said Max.

"One of these days I’ll invite you to my apartment and you’ll realize I’m not underestimating my collection...". Spencer paused for a few seconds, hoping it didn't sound like an overly explicit invitation. Apparently Max took it very naturally.

"It would be nice to see that, although I could bet you do have a giant collection..." Max laughed. "Well, what else…?"

“Uhm… I like to watch old movies sitting on my couch eating popcorn. I also like listening classical music on an old record player I have… sometimes I write little case reports for my classes…” said Spencer.

"No offense, I think I'm beginning to understand why your therapist told you to go out more..." Max said with some caution to try not to make Spencer feel bad. He just laughed out loud.

"The truth is… I like my place. I think it has everything I need for when I'm not at work. Even though…"

"Even though …?"

"Even though I think walking in the park and talking with you could also be one of my new favorite activities...". Max couldn't help but blush, knowing perfectly well what he was referring to.

Spencer couldn't help saying it either and though it might seem like he was flirting but he really was convinced of how good a practice like that had done to him. Gratitude perhaps? He thought was clear it could be more than that. Her eyes captivated him. When she speak… the passion was clear in her words. She looked secure, defined in life... he, in other hand, being the genius he was, felt he lacked. "Had she suffered much in her life?" he wondered. He hoped not. He inadvertently felt a need to protect her but he was not sure why,  because it was clear that she didn’t need any man to protect her.

Time had gone by too quickly for both of them. When Max saw the clock on her phone realized if she did not leave at that moment would be late for class.

"I have to go now, the next class is in 20 minutes" said Max. Spencer gestured for the waiter to bring over the check. "Thank you so much for lunch" she said while was getting up from the chair. Spencer rushed to stop her.

"Wait, I'll go with you…"

"Spencer, it is not necessary, I can take a cab outside…".

“Let me go with you. I brought you here. At least let me take you back” said Spencer.

"I'm not going to get lost, if that's what worries you…" Max said jokingly. The waiter had brought the check and Spencer was paying with his credit card.

"I know you're not going to get lost... it's just that...". The waiter finished processing the payment and left. "Let me go with you".

Max didn’t understand why so much insistence by him. Under normal conditions she would believe this excess courtesy was a typical tactic of male conquest, but in Spencer it’s looks like very natural. There was no way he was faking politeness.

She finally agreed to have Spencer's company in a cab. The funny thing is throughout the trip none of them spoke. They only gave each other furtive glances hoping the other didn't notice. Despite that the trip wasn't awkward. Apparently they liked having that kind of company even if they was silent. Max felt in comfort with his company even though they knew each other so little. She wondered if he would feel something similar.

When they got back to school it was almost 3:00 p.m. They both got out of the taxi. Max was in a bit of a rush, but she didn't want to be rude, so she let Spencer walked with her to the entrance. They stood looking at each other for a few seconds.

"Thanks for the lunch. I had a great time with you today… and the risotto was totally worth it” said Max smiling.

"No problem. Thanks to you for stand me all these hours…” Spencer said jokingly. Max couldn't help but patting his arm.

"Hey! Don't say that. You don't seem to be torture after all”. Max laughed.

"Good to know it... Ok, I'm leaving so you can go to your class”. Spencer moved closer so he could kiss her on the cheek. He hesitated a few seconds, but when he saw that she was approaching him, he decide to do it. It was a short but very warm and pleasant kiss.

Goodbye Spencer. Take care” Max said before turning and come into the school. It was just time to start her class.

"You too..." Spencer managed to say, although in a voice so soft and low that hardly she heard it. Then, with his hands in his pockets, he began his walk in search of a cab to return to work.

--------------------

[ncertsolutionsbooks]

NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals Ex 8.2

NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals Ex 8.2

NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals

Get Free NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals Ex 8.2 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Application of Integrals Exercise 8.2 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers.…

[valpy-blog]

NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals Ex 8.2

NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals Ex 8.2

NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals

Get Free NCERT Solutions for Class 12 Maths Chapter 8 Application of Integrals Ex 8.2 PDF in Hindi and English Medium. Sets Class 12 Maths NCERT Solutions are extremely helpful while doing your homework. Application of Integrals Exercise 8.2 Class 12 Maths NCERT Solutions were prepared by Experienced LearnCBSE.in Teachers.…