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anonymous

  • one year ago

Need Help! Evaluate the Definite integral with U- substitution.

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  1. anonymous
    • one year ago
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    |dw:1439090251436:dw|

  2. ganeshie8
    • one year ago
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    look up the derivative of \(\sin^{-1}x\)

  3. ganeshie8
    • one year ago
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    substitute \(u=\sin^{-1}x\) then \(du=?\)

  4. anonymous
    • one year ago
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    I'm confused on how you got U to be sin-1x, how did you know it was that?

  5. anonymous
    • one year ago
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    I thought U was supposed what's inside roots or paretheses, such as 1-x^2

  6. ganeshie8
    • one year ago
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    do you know what the derivative of \(\sin^{-1}x\) is ?

  7. anonymous
    • one year ago
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    Yea...it's 1/srt 1-x^2 * dx

  8. ganeshie8
    • one year ago
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    that means substituting \(u = \sin^{-1}x\) simplifies the integrand, because \(du = \dfrac{1}{\sqrt{1-x^2}}dx\) that entire radical mess can be replaced by \(du\)

  9. anonymous
    • one year ago
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    So I can't always assume that whats inside the parentheses or roots is always going to be substituted as U, right?

  10. ganeshie8
    • one year ago
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    |dw:1439090695448:dw|

  11. anonymous
    • one year ago
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    I get that!

  12. ganeshie8
    • one year ago
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    No, with integrals there are no "strict" rules, you will have to do some guessing with each and every integral problem

  13. anonymous
    • one year ago
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    okay thanks!

  14. ganeshie8
    • one year ago
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    Also, don't forget to change the bounds accordingly

  15. ganeshie8
    • one year ago
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    |dw:1439090839359:dw|

  16. anonymous
    • one year ago
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    Why would I need to change the bounds?

  17. ganeshie8
    • one year ago
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    because with u-substitution you're changing the variable of integration, so the bounds also change accordingly

  18. ganeshie8
    • one year ago
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    |dw:1439090975245:dw|

  19. ganeshie8
    • one year ago
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    those bounds refer to \(x\), they don't belong to \(u\)

  20. anonymous
    • one year ago
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    |dw:1439091066367:dw|

  21. anonymous
    • one year ago
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    |dw:1439091093678:dw|

  22. ganeshie8
    • one year ago
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    no wait, whats the integrand right after substituting ?

  23. anonymous
    • one year ago
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    |dw:1439091171917:dw|

  24. ganeshie8
    • one year ago
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    Yes, whats antiderivative of \(u\) with respective to \(u\) ?

  25. anonymous
    • one year ago
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    U(x)?

  26. ganeshie8
    • one year ago
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    nope, whats antiderivative of \(x\) with respect to \(x\) ?

  27. anonymous
    • one year ago
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    ohh....so would it be U^2/2

  28. ganeshie8
    • one year ago
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    Yes

  29. anonymous
    • one year ago
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    but isnt U supposed to be thought of a constant, therefore it's antiderivative should be U of x?

  30. ganeshie8
    • one year ago
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    \(u\) is not constant, it is the variable that has replaced \(\sin^{-1}x\)

  31. anonymous
    • one year ago
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    |dw:1439091490507:dw|

  32. ganeshie8
    • one year ago
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    Looks good!

  33. anonymous
    • one year ago
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    So do I get a decimal answer when I evaluate the integral?

  34. ganeshie8
    • one year ago
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    plugin the bounds and see

  35. anonymous
    • one year ago
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    Okay so I got (sin-1(1/2))^2/2? So do I box in that as the answer or its decimal...which is .137077

  36. ganeshie8
    • one year ago
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    recall that \(\sin(\pi/6)=1/2\)

  37. Ac3
    • one year ago
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    look at the directions in the question

  38. anonymous
    • one year ago
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    The directions in the questions just say evaluate the integral.

  39. Ac3
    • one year ago
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    if it asks for EXACT answers which most professors want then you give them that

  40. Ac3
    • one year ago
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    answer should be pi/6

  41. Ac3
    • one year ago
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    never do decimal unless it specifically tells you to. We know that arcsin of 1/2 is pi/6 because of the unit circle

  42. anonymous
    • one year ago
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    ganeshie, is the answer (pi/6)^2/2?

  43. Ac3
    • one year ago
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    my bad pi/12

  44. Ac3
    • one year ago
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    wow i'm terrible today

  45. anonymous
    • one year ago
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    hmm i got 5/2pi

  46. ganeshie8
    • one year ago
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    haha thats not it

  47. anonymous
    • one year ago
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    lol i need help then..

  48. Ac3
    • one year ago
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    the answer i gave is wrong it should be pi/72 i'll show simplification right now one sec.

  49. ganeshie8
    • one year ago
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    nope thats still wrong

  50. anonymous
    • one year ago
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    ah i got it pi^2/72

  51. ganeshie8
    • one year ago
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    looks good! :)

  52. anonymous
    • one year ago
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    Yes thats what I got too saseal!

  53. Ac3
    • one year ago
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    |dw:1439091910076:dw|

  54. Ac3
    • one year ago
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    i meant to say pi^2/72 originally

  55. ganeshie8
    • one year ago
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    yes leave it as \(\dfrac{\pi^2}{72}\) do not convert to decimals unless asked to do so

  56. anonymous
    • one year ago
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    Okay, cool, thanks ganeshie!

  57. Ac3
    • one year ago
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    Usually with u-substitution the rule of thumb is to try and get something to go away with your Du. That's where knowing you derivatives really well comes in to play. When in doubt just guess and check.

  58. anonymous
    • one year ago
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    Thanks Ac3!

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