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cherrilyn

  • 5 years ago

evaluate the integral

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  1. cherrilyn
    • 5 years ago
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    \[\int\limits_{?}^{?}(x ^{2} + 3x -44)dx/(x+3)(x+5)(3x-2)\]

  2. amistre64
    • 5 years ago
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    partial fraction decomp it looks like; unless we can factor the top to cancel some bottoms

  3. amistre64
    • 5 years ago
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    x^2 +3x -44 = A(x+5)(3x-2) ; x = -3 x^2 +3x -44 = B(x+3)(3x-2) ; x = -5 x^2 +3x -44 = C(x+3)(x+5) ; x = 2/3

  4. cherrilyn
    • 5 years ago
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    woohoo thats what I got. but now how do I find the actual values of the letters?

  5. amistre64
    • 5 years ago
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    -44 = A(2)(-11) -44/-22=A A = 2 ---------------------- 25 -59 = B(-2)(-17) -34 = B(34) B = -1 x^2 +3x -44 = C(x+3)(x+5) ; x = 2/3

  6. amistre64
    • 5 years ago
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    when x = a number that zeros the other 2; use it and solve for that letter

  7. cherrilyn
    • 5 years ago
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    but sometimes it leaves behind more than one variable

  8. amistre64
    • 5 years ago
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    x^2 +3x -44 = C(x+3)(x+5) ; x = 2/3 (2/3)^2 +3(2/3) - 44 = C(2/3 +3)(2/3 +5)

  9. amistre64
    • 5 years ago
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    sometimes it does, but use it first to get some solids :) with linears on the bottom, its not a problem

  10. cherrilyn
    • 5 years ago
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    okay. for the last equation I got 4/9 - 42 = c(2)(2/3 + 5)

  11. amistre64
    • 5 years ago
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    (4/9) +2 - 44 = C(11/3)(17/3) (4/9) - 42 = C(11/3)(17/3)

  12. amistre64
    • 5 years ago
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    -374/9 = C(187/9) C = -374/189 reduce as wanted

  13. cherrilyn
    • 5 years ago
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    wow how do you work so fast?

  14. amistre64
    • 5 years ago
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    i use my head and mess up alot lol

  15. cherrilyn
    • 5 years ago
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    lol oh... do you think you can help me with another similar problem?

  16. amistre64
    • 5 years ago
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    perhaps; but let me get thru this one first :)

  17. cherrilyn
    • 5 years ago
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    okay! thanks for all of your help :)

  18. amistre64
    • 5 years ago
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    we did C correctly right? that monster fraction scares me into thinking I messed it up lol

  19. amistre64
    • 5 years ago
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    i typoed the 187 into a 189 ;)

  20. amistre64
    • 5 years ago
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    C = 2 :)

  21. amistre64
    • 5 years ago
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    -2

  22. cherrilyn
    • 5 years ago
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    wait, in this problem.. there is no Dx+E?

  23. amistre64
    • 5 years ago
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    so our integral becomes: [S] 2/(x+3) - 1/(x+5) - 2/(3x-2) dx

  24. amistre64
    • 5 years ago
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    the bottom is already factored for us into linear forms; linear form dont need the Dx+E form

  25. cherrilyn
    • 5 years ago
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    ahh okay! Yayyyy. :D

  26. amistre64
    • 5 years ago
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    if we had an irreducible quadratic in the denom; then we would use the Dx+E form above it to aid us

  27. cherrilyn
    • 5 years ago
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    like.. (x-1)^3?

  28. amistre64
    • 5 years ago
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    nope; that is a multiple of a linear and still only needs a A B C single atop it

  29. cherrilyn
    • 5 years ago
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    really? how would it look.. A/x-1 + B/x-1 + c/x-1?

  30. amistre64
    • 5 years ago
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    x^2 + 10 is irreducible quadratic; cant be factored to real linears

  31. amistre64
    • 5 years ago
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    A/(x-1) + B/(x-1)^2 + C/(x-1)^3

  32. cherrilyn
    • 5 years ago
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    ohh.. hmm . and that would be... Ax+B/x^2+10?

  33. amistre64
    • 5 years ago
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    our integral has become: 2ln(x+3) - ln(x+5) -2ln(3x-2) +C

  34. cherrilyn
    • 5 years ago
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    thats the final answer

  35. amistre64
    • 5 years ago
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    thats the form we would use above it yes :)

  36. amistre64
    • 5 years ago
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    we could probably turn this into one ln(yadayada)+C if we want, but this is good :)

  37. cherrilyn
    • 5 years ago
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    wow thank you!!!!

  38. amistre64
    • 5 years ago
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    youre welcome :)

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