anonymous
  • anonymous
Please explain integration step.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
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.
amistre64
  • amistre64
expand the bottom to determine the divisor
amistre64
  • amistre64
and then the results splits apart into the quotient

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amistre64
  • amistre64
http://www.wolframalpha.com/input/?i=x%5E3%2F%28x-2%29%5E2
amistre64
  • amistre64
that has show steps in the division process
amistre64
  • amistre64
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
amistre64
  • amistre64
8(x-2) --------- (x-2)(x-2)
amistre64
  • amistre64
well, something like that lol; 16-4 = 12 not 8
amistre64
  • amistre64
12x-16 = 2(6x-8) hmm got me, maybe there are 2 ways to express it? might have to work at simplifing further
dumbcow
  • dumbcow
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
amistre64
  • amistre64
oh, they got the remainder and decomped it?
dumbcow
  • dumbcow
yeah
anonymous
  • anonymous
Thank you!
anonymous
  • anonymous
I think I can handle it from there. :)

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