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anonymous

  • 4 years ago

Please explain integration step.

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  1. anonymous
    • 4 years ago
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    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.

  2. amistre64
    • 4 years ago
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    expand the bottom to determine the divisor

  3. amistre64
    • 4 years ago
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    and then the results splits apart into the quotient

  4. amistre64
    • 4 years ago
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    http://www.wolframalpha.com/input/?i=x%5E3%2F%28x-2%29%5E2

  5. amistre64
    • 4 years ago
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    that has show steps in the division process

  6. amistre64
    • 4 years ago
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    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

  7. amistre64
    • 4 years ago
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    8(x-2) --------- (x-2)(x-2)

  8. amistre64
    • 4 years ago
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    well, something like that lol; 16-4 = 12 not 8

  9. amistre64
    • 4 years ago
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    12x-16 = 2(6x-8) hmm got me, maybe there are 2 ways to express it? might have to work at simplifing further

  10. dumbcow
    • 4 years ago
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    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

  11. amistre64
    • 4 years ago
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    oh, they got the remainder and decomped it?

  12. dumbcow
    • 4 years ago
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    yeah

  13. anonymous
    • 4 years ago
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    Thank you!

  14. anonymous
    • 4 years ago
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    I think I can handle it from there. :)

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