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anonymous

  • 5 years ago

can someone help me solve an integral?

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  1. anonymous
    • 5 years ago
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    |dw:1326769120780:dw|

  2. Akshay_Budhkar
    • 5 years ago
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    the thing that immediately comes to my mind is use the \[\int\limits_{}^{} uvdx\] formula twice

  3. anonymous
    • 5 years ago
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    ok in the midst of doing that

  4. Akshay_Budhkar
    • 5 years ago
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    but that will be a lengthy business, must have a shorter route

  5. anonymous
    • 5 years ago
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    so what do u suggest?

  6. anonymous
    • 5 years ago
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    cos(x)= Re[e^(i x)] x= e^(ln x) \[\int e^{ln(x)} e^x e^{i x} dx\]

  7. anonymous
    • 5 years ago
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    \[\int e^{ln(x)+x+i x} dx\]

  8. Akshay_Budhkar
    • 5 years ago
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    Complex numbers!! y didnt i think of that!

  9. anonymous
    • 5 years ago
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    huh what is e^ix?

  10. anonymous
    • 5 years ago
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    cos(x)= Re[e^(i x)] what is this step?

  11. anonymous
    • 5 years ago
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    Using integration by parts three times to solve it isn't that bad either. It would take a while to type out all of the equations, but if you use 1. u=xe^x, dv=cos(x), then 2. u=xe^x, dv=sin(x), then 3. u=e^x, dv=cos(x) over your three stages, you will get the answer of \[\int xe^x\cos x dx = \frac{1}{2}xe^x(\sin x + \cos x) - \frac{1}{2}e^x\sin x + C\]

  12. anonymous
    • 5 years ago
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    Eh. I would hesitate using complex numbers if you don't know how, you can get yourself into trouble.... using the natural extension of integration by parts to three functions is simple and straightforward, and makes sense already.

  13. Akshay_Budhkar
    • 5 years ago
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    @imperialist that will be a bit of lengthy business though, if she knows complex numbers it will be faster

  14. anonymous
    • 5 years ago
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    @Jemurray3, I agree. It would take far longer to explain why you can use complex numbers here (and even longer to understand it!) than 3 applications of integration by parts.

  15. anonymous
    • 5 years ago
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    Thanks guys :D

  16. anonymous
    • 5 years ago
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    I am just gonna wait to see what myin has to say for herself

  17. myininaya
    • 5 years ago
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    First lets evaluate: \[\int\limits_{}^{}e^x \cos(x) dx\] \[=e^x \cos(x)-\int\limits_{}^{}e^x(-\sin(x)) dx=e^x \cos(x)+\int\limits_{}^{}e^x \sin(x) dx\] \[=e^x \cos(x)+(e^x \sin(x)-\int\limits_{}^{}e^x \cos(x) dx)\] So we have \[\int\limits_{}^{}e^x \cos(x) dx=e^x \cos(x)+(e^x \sin(x)-\int\limits_{}^{}e^x \cos(x) dx)\] So we shall solve this for: \[\int\limits_{}^{}e^x \cos(x) dx\] So we have \[\int\limits_{}^{}e^x \cos(x) dx=\frac{1}{2}e^x \cos(x)+\frac{1}{2}e^{x}\sin(x)+c_1\] ok now let's look back at \[\int\limits_{}^{} x e^x \cos(x) dx\] Apply integration by parts again! :) \[=(\frac{1}{2}e^x \cos(x)+\frac{1}{2}e^x \sin(x))x-\frac{1}{2}\int\limits_{}^{}(e^x \cos(x)+e^x \sin(x)) dx\] So we still need to find \[\int\limits_{}^{}e^x \sin(x) dx\] So we will need integration by parts again \[=e^x \sin(x)-\int\limits_{}^{}e^x \cos(x) dx\] \[=e^x \sin(x)-(e^x \cos(x)-\int\limits_{}^{}e^x (-\sin(x)) dx)\] So we need to solve this for: \[\int\limits_{}^{}e^x \sin(x) dx\] \[\int\limits_{}^{}e^x \sin(x) dx=\frac{1}{2} e^x \sin(x)-\frac{1}{2}e^x \cos(x)+ c_2\] So we need to put all of this together! lol

  18. myininaya
    • 5 years ago
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    \[\int\limits_{}^{}x e^x \cos(x) dx=\frac{1}{2}x e^x \cos(x)+\frac{1}{2}x e^x \sin(x)-\frac{1}{2}(\frac{1}{2}e^x \cos(x)+\frac{1}{2}e^x \sin(x)+\] \[\frac{1}{2}e^x \sin(x)-\frac{1}{2}e^x \cos(x))+C\] after adding up all the constants I get C lol because the sum of some constants is still constant I think I got every part in there

  19. anonymous
    • 5 years ago
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    LOL WOW

  20. anonymous
    • 5 years ago
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    Thanks myin but u did it differently then what the others suggested

  21. anonymous
    • 5 years ago
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    I wish I could give you a dozen medals myininaya for typing all of that up! :)

  22. Akshay_Budhkar
    • 5 years ago
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    i tell her that for every answer! she is more than cool! :D

  23. Akshay_Budhkar
    • 5 years ago
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    just tell ur kids not to trouble me @ myin :P :P

  24. anonymous
    • 5 years ago
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    Thanks myin imperialist and jemurray:D I will have to review th steps now :D

  25. myininaya
    • 5 years ago
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    Great question.

  26. myininaya
    • 5 years ago
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    You guys are really sweet! :)

  27. Akshay_Budhkar
    • 5 years ago
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    ur kids are sweet too :D

  28. Akshay_Budhkar
    • 5 years ago
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    so are you :D

  29. anonymous
    • 5 years ago
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    Oh myin u deserve the medal not me LOL

  30. myininaya
    • 5 years ago
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    I don't have kids lol

  31. myininaya
    • 5 years ago
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    you have kids not me lol

  32. anonymous
    • 5 years ago
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    Akshay is like1o years younger than you LOL

  33. Akshay_Budhkar
    • 5 years ago
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    yea i told her that she doesnt get me! her kidds come to my bakery and dont pay me

  34. anonymous
    • 5 years ago
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    LOL and eat ur trash

  35. Akshay_Budhkar
    • 5 years ago
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    LOL junk to be precise

  36. myininaya
    • 5 years ago
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    lol

  37. Akshay_Budhkar
    • 5 years ago
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    myin is blushing when i told her she has kids :P

  38. anonymous
    • 5 years ago
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    lol she is dating satellite LOL hehe

  39. Akshay_Budhkar
    • 5 years ago
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    !

  40. Akshay_Budhkar
    • 5 years ago
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    satellite is 40 years old!

  41. myininaya
    • 5 years ago
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    You guys are crazy. I never met satellite in person.

  42. anonymous
    • 5 years ago
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    lol i was teasing

  43. myininaya
    • 5 years ago
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    i know

  44. Akshay_Budhkar
    • 5 years ago
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    rld study dont tease :P :P

  45. anonymous
    • 5 years ago
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    yup got loads of work to do. Thanks myin U r awesome

  46. Akshay_Budhkar
    • 5 years ago
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    what about me?

  47. anonymous
    • 5 years ago
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    LOL dont even have to say that

  48. Akshay_Budhkar
    • 5 years ago
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    yea yea~ :P i m so cool :P

  49. anonymous
    • 5 years ago
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    if u live in my city = cool

  50. Akshay_Budhkar
    • 5 years ago
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    later=D

  51. anonymous
    • 5 years ago
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    ya bye guys

  52. anonymous
    • 5 years ago
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    myin helped me solve it but i know u like integrals. The second I posted it u disappeared LOL Too bad

  53. anonymous
    • 5 years ago
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    My bad, I'll help on the next one.

  54. anonymous
    • 5 years ago
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    ya but now i am doing stupis things. I gotta refer to the tables of integrands and solve accordingly. Soon I will be solving using trig subsitution so i will be back :D

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