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anonymous

  • one year ago

how to do this integral?

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  1. anonymous
    • one year ago
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    \[\int\limits_{}\frac{ x^2 + 1 }{ (x-3)(x-2)^2}dx\]

  2. misty1212
    • one year ago
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    HI!!

  3. anonymous
    • one year ago
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    hello!!

  4. misty1212
    • one year ago
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    how are you at partial fractions?

  5. anonymous
    • one year ago
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    i understand it a bit (split into parts, right? A/B?) but don't know quite how to apply it!

  6. misty1212
    • one year ago
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    it is going to suck want to do it step by step, or cheat?

  7. anonymous
    • one year ago
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    can you show me how to do it step by step?

  8. misty1212
    • one year ago
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    \[\frac{A}{x-3}+\frac{B}{x-2}+\frac{C}{(x-2)^2}\]

  9. anonymous
    • one year ago
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    okay! what next? :O

  10. misty1212
    • one year ago
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    that means \[A(x-2)^2+B(x-3)(x-2)+C(x-3)=x^2+1\]

  11. misty1212
    • one year ago
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    let me know if that is clear, then we can continue

  12. anonymous
    • one year ago
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    yes:)

  13. anonymous
    • one year ago
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    it is clear:) what happens next?

  14. misty1212
    • one year ago
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    we can get C right away by putting \(x=2\) because the A and B will drop out

  15. misty1212
    • one year ago
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    if you put \(x=2\) you get \[C(2-3)=2^2+1\] or \[-C=5\] making \(C=-5\)

  16. misty1212
    • one year ago
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    got that?

  17. anonymous
    • one year ago
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    yes:) sorry, OS connection was very poor!! what happens next?

  18. IrishBoy123
    • one year ago
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    get A by setting x = 3 same technique

  19. anonymous
    • one year ago
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    okay, so then i get 3^2 + 1 = 10 ?

  20. anonymous
    • one year ago
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    A=10?

  21. IrishBoy123
    • one year ago
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    \[A(3-2)^2+B(3-3)(3-2)+C(3-3)=3^2+1\] \[\checkmark\]

  22. anonymous
    • one year ago
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    ok!! what do i do next?

  23. IrishBoy123
    • one year ago
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    \[10(x-2)^2+B(x-3)(x-2)-5(x-3)=x^2+1\] you need B so set x to anything really. but let's just go with x = 0.

  24. anonymous
    • one year ago
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    okay! so B=1?

  25. IrishBoy123
    • one year ago
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    sure? 10(4) + B(-3)(-2) - 5(-3) = 1 6B = (1 -15 -40)????

  26. anonymous
    • one year ago
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    ohh oops!! it equals -55!

  27. anonymous
    • one year ago
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    -0.1111 = B?

  28. anonymous
    • one year ago
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    @IrishBoy123 i have to leave for an appointment now, but will be back in a few hours!! is it okay if we continue then, or if i come back and see what you say?

  29. IrishBoy123
    • one year ago
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    \[\color{red}6B = -5 \color{red}4\]

  30. IrishBoy123
    • one year ago
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    sure

  31. anonymous
    • one year ago
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    @IrishBoy123 I'm back!! can you please help me explain the rest?

  32. anonymous
    • one year ago
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    so B = -9 right? what happens next?

  33. IrishBoy123
    • one year ago
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    let's agree where we got to first

  34. IrishBoy123
    • one year ago
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    \[\frac{10}{x-3}-\frac{9}{x-2}-\frac{5}{(x-2)^2}\] are we sure that's a good summary?

  35. anonymous
    • one year ago
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    okay! so A=10, B=-9 and C=-5 ?

  36. IrishBoy123
    • one year ago
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    now you must do \[\int \frac{10}{x-3}dx- \int \frac{9}{x-2}dx-\int\frac{5}{(x-2)^2} dx\] 3 separate integrals

  37. anonymous
    • one year ago
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    ooh okay, how do i start off?

  38. IrishBoy123
    • one year ago
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    first one \[\int \frac{10}{x-3}dx\]

  39. anonymous
    • one year ago
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    okay, so we get 10lnx-3 ?

  40. IrishBoy123
    • one year ago
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    y second?

  41. anonymous
    • one year ago
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    so it is not that?

  42. IrishBoy123
    • one year ago
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    \[\int \frac{9}{x-2}dx\]

  43. anonymous
    • one year ago
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    or it is?

  44. IrishBoy123
    • one year ago
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    first part is right, do the second now

  45. anonymous
    • one year ago
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    ohhh i see now and so now second we get 9lnx-2 ?

  46. anonymous
    • one year ago
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    and for the third, we get -5/(x-2) ?

  47. IrishBoy123
    • one year ago
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    yes. just make a note that we need to put the correct signs between the integrals righ at the end as it is ∫dx - ∫dx - ∫dx last one \[\int\frac{5}{(x-2)^2} dx\]

  48. IrishBoy123
    • one year ago
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    yes put it all together now

  49. IrishBoy123
    • one year ago
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    don't bother writing it out if you are happy with it all

  50. anonymous
    • one year ago
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    \[\int\limits_{}10lnx-3 dx - \int\limits_{}9lnx-2dx + \int\limits_{} \frac{ 5 }{ x-2 }dx\]

  51. anonymous
    • one year ago
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    like that?

  52. IrishBoy123
    • one year ago
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    no, drop all the integration signs now that you have integrated so \[10 \ln(x-3) - 9\ln(x-2) + \frac{ 5 }{ x-2 }\]

  53. IrishBoy123
    • one year ago
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    OK?

  54. anonymous
    • one year ago
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    ohh okay!! what now? :)

  55. IrishBoy123
    • one year ago
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    i think the fat lady is now singing.

  56. anonymous
    • one year ago
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    hahaha ohh so we are done?

  57. anonymous
    • one year ago
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    @IrishBoy123 ?

  58. IrishBoy123
    • one year ago
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    OMG!!!! we forgot something....

  59. anonymous
    • one year ago
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    what?? :O

  60. IrishBoy123
    • one year ago
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    the integration constant!!

  61. IrishBoy123
    • one year ago
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    how embarassing after all that :-))

  62. anonymous
    • one year ago
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    ohh + C ?

  63. IrishBoy123
    • one year ago
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    indeed!

  64. anonymous
    • one year ago
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    ohh okie!! yay!! thank you so much!!

  65. IrishBoy123
    • one year ago
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    you and misty did all the heavy lifting, we just finished it off:) good night!

  66. anonymous
    • one year ago
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    good night!:D

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