Question

Please explain integration step.

Answer 1

I needed to integrate x^3/(x-2)^2.

So, I did long division and got x + 4 as the quotient and 12x - 16 as th remainder.

So, to get the answer, I was going to take the sum of integrals : ∫x + ∫4 + ∫(12/(x-2)^2) - ∫(16/(x-2)^2).

But multiple people have told me that for the last two terms, I should have ∫(12/(x-2)) - ∫(8/(x-2)^2) instead.

They said they got it through long division, but I still don't understand how.

Please explain. Thanks.

Answer 2

expand the bottom to determine the divisor

Answer 3

and then the results splits apart into the quotient

Answer 4

http://www.wolframalpha.com/input/?i=x%5E3%2F%28x-2%29%5E2

Answer 5

that has show steps in the division process

Answer 6

x+4 + (8x-16)/D
---------------
x^2-4x+4 ) x^3
(x^3-4x^2+4x)
-------------
4x^2-4x
(4x^2-16x+16)
-------------
8x - 16

and the Remainder might be able to factor out with the D

Answer 7

8(x-2)
---------
(x-2)(x-2)

Answer 8

well, something like that lol; 16-4 = 12 not 8

Answer 9

12x-16 = 2(6x-8) hmm

got me, maybe there are 2 ways to express it? might have to work at simplifing further

Answer 10

I don't see why it matters, you get the same result either way.

but basically they did partial fractions backwards
12x -16/(x-2)^2 = A/(x-2) + B/(x-2)^2

A(x-2) + B = 12x -16
Ax -2A +B = 12x -16
A=12
-2(12)+B = -16
B = 8

Answer 11

oh, they got the remainder and decomped it?

Answer 12

yeah

Answer 13

Thank you!

Answer 14

I think I can handle it from there. :)