#thebigbangingboom

[maths-help]

i'm here to ask for help! if you have any idea how to explain solving integrals by partial fractions i will love you forever.

Okay, so partial fraction decomposition:

The idea behind this technique is that we know how to integrate things in the form

 pretty easily. (Namely, you get ) So we want to use this fact to help evaluate integrals of the form 

 i.e. integrals with a polynomial in the denominator.

To use partial fraction decomposition, your polynomial must be factorable into linear factors. Let's do an example. Suppose you want to evaluate

  What we're going to do is break up 

 as the sum of two fractions. (In general, if your polynomial is a degree n polynomial, you will have n terms in your sum.) 

Notice if we take some multiple of  and add it to some multiple of  the sum is going to have (x+3)(2x-5) in the denominator! This is the key idea that makes partial fraction decomposition work.

So we set 

 This holds true for all x, so in particular, if we set x = -3, we get

 or 1 = -11A. This gives you A = -1/11. Similarly, letting x = 5/2, we get B = 2/11.

So now we know 

 What does this mean? 

We know integrals are additive, i.e.  So if we want to evaluate 

 all you have to do is evaluate

  But we can do that! (See way up top.)

There's something slightly subtler that happens when your polynomial has repeated factors, though. Consider taking the partial fraction decomposition of  You can't just take  because you won't get (x-1)^2 in the denominator. Instead, what you're going to have to do is take 

 In general, if your polynomial has the factor (x-a)^n, you want to decompose it into 

  and all these terms should be integrable because you have functions in the form (deg n-1 polynomial)/(deg n polynomial). 

Hope that helped!

- H2

[dersneakerking]

Ich glaube dass wird nicht passieren. Genau deswegen wird es geschehen!

The big bang Theory

[gaydestiny]

thebigbangingboom replied to your post: why is popular music so fucking bad 

the Beatles are popular and they aren’t bad…

I was thinking more like Lil Wayne and most of the artists who performed at the VMAs 

[amicusbalderdash]

thebigbangingboom replied to your post: My decision to join my parents at church today...

do parents have phases too? or does that only apply for their kids…? idk i was gonna say don’t worry it’s just a phase…

Yeah, phases happen all throughout life :) Unfortunately it's a pretty concrete opinion of theirs that 'it's a sin/unnatural bla bla, holier than thou etc.' :/