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RedPrince

What is the integral of ∫(cos x)^2.sqrt{sin x}dx

  • one year ago
  • one year ago

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  1. lgbasallote
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    \((\cos x)^2 = \cos^2 x = 1 - \sin ^2 x\) i think this can be applied? but seems no :/

    • one year ago
  2. RedPrince
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    @lgbasallote It is not working. I already checked it.....!!

    • one year ago
  3. simran
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    |dw:1336216786369:dw|

    • one year ago
  4. simran
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    is dis a question?

    • one year ago
  5. estudier
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    Maybe it is doable with the Weierstrass substitution?

    • one year ago
  6. RedPrince
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    @simran yes it is aquestion

    • one year ago
  7. RedPrince
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    @estudier what you want to say.... explain please

    • one year ago
  8. RedPrince
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    |dw:1336347046770:dw|

    • one year ago
  9. estudier
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    Weierstrass sub sin x = 2t/(1+t^2) cos x = (1-t^2)/(1+t^2) dx = 2 dt/(1+t^2)

    • one year ago
  10. badi
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    \[2(7\sin ^{3/2}x-3\sin ^{7/2})/21\]

    • one year ago
  11. RedPrince
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    \[\int\limits (cosx)^2\sqrt{sinx }dx= \int\limits (1-t^2)/(1+t^2) 2/(1+t^2) dx\] @estudier this is you want to say!

    • one year ago
  12. RedPrince
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    @badi explain please

    • one year ago
  13. shivam_bhalla
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    One tip @RedPrince To write as fractions in latex Use the folowing syntax frac{numerator}{denominator}

    • one year ago
  14. RedPrince
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    @shivam_bhalla sorry I am a new user on this site! I will keep it in my mind in future. thanks for tip

    • one year ago
  15. shivam_bhalla
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    No problem. :). trying to solve your problem :D

    • one year ago
  16. RedPrince
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    @shivam_bhalla I tried my best but failed!

    • one year ago
  17. shivam_bhalla
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    @RedPrince , the problem is solved. First write \[\cos^2 x = \frac{(1+\cos(2x))}{2}\] We get \[\frac{1}{2}(\int\limits_{}^{}{\sqrt{sinx}dx} +\int\limits_{}^{}{\cos(2x){sinx}dx} )\] Now we integate them separately 1)\[\int\limits_{}^{}{\sqrt{sinx}dx} \] Take \[\sin x = t^2\] Therefore \[cosx dx=2tdt\] \[dx =\frac{ 2t}{cosx}\] Now write \[cosx=\sqrt{1-\sin^2 x}=\sqrt{1-t^4}\] Finally you get \[\int\limits_{}^{} \sqrt{sinx}dx = \int\limits_{}^{}\frac{2tdt}{\sqrt{1-t^4}}\] Now you should be able to integrate this. 2)\[\int\limits_{}^{}(\cos(2x))sinxdx=\int\limits_{}^{}(2\cos^2 x-1)sinxdx\] Now take p=cos x dp= -sinx dx \[-\int\limits_{}^{}{(2t^2-1)}dt\] Now you can integrate this too. Integrate both and finally subtitute t an p in terms of x and simplify. It is pretty long but the only solution which comes to my mind

    • one year ago
  18. shivam_bhalla
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    Oops in part 2 of the problem, it should be \[-\int\limits\limits_{}^{}{(2p^2-1)}dp\]

    • one year ago
  19. RedPrince
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    @shivam_bhalla Ok w8 I try it

    • one year ago
  20. experimentX
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    looks like we need something called elliptic integrals http://www.wolframalpha.com/input/?i=integrate+sqrt%28sinx%29

    • one year ago
  21. shivam_bhalla
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    @experimentX , I am getting the answer with my method :D

    • one year ago
  22. experimentX
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    \( \int \cos 2x \sqrt{\sin x} dx \) <---- i guess there was error here in your method.

    • one year ago
  23. RedPrince
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    @shivam_bhalla I think you did a mistake! there is: \[\cos^2(2x).\sqrt{sinx}\] instead of \[\cos^2(2x).sinx\]

    • one year ago
  24. shivam_bhalla
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    Yes. Sorry for the mistake in part 2 of the problem. let me correct it and tell you

    • one year ago
  25. estudier
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    Did anybody try Weierstrass substitution?

    • one year ago
  26. RedPrince
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    @shivam_bhalla it's ok!!! I am waiting and try Weierstrass substitution as @estudier is said

    • one year ago
  27. RedPrince
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    I try Weierstrass substitution

    • one year ago
  28. shivam_bhalla
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    @estudier , I tried but getting some unsolvable integral. why don't you try and see :D

    • one year ago
  29. experimentX
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    something doesn't seem right http://www.wolframalpha.com/input/?i=integrate+2x%2Fsqrt%281+-+x^4%29

    • one year ago
  30. shivam_bhalla
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    @experimentX , that is the solution for part 1. Now I am stuck at part 2 of the integration

    • one year ago
  31. shivam_bhalla
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    @experimentX , your initial doubt is right. The solution for 2nd part of intergal is http://www.wolframalpha.com/input/?i=%E2%88%AB%28cos%282x%29%29%28sqrt%28sin%28x%29%29%29dx

    • one year ago
  32. RedPrince
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    @shivam_bhalla I already said that I tried my best but failed I black 10 papers for this question! yes @experimentX this is solution for the first part! I continue to solve this question

    • one year ago
  33. shivam_bhalla
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    @experimentX ,Looks like we need to learn elliptical integral.

    • one year ago
  34. experimentX
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    @shivam_bhalla i think you need to sleep :D :D http://www.wolframalpha.com/input/?i=integrate+2x^2%2Fsqrt%281+-+x^4%29 you missed t from \( \sqrt{\sin x}\) in the first part

    • one year ago
  35. shivam_bhalla
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    LOL. looks like I really need to sleep now. Sorry @RedPrince .@experimentX .

    • one year ago
  36. RedPrince
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    @shivam_bhalla it's ok dear text my friend and he reply that he solve it! when I get answer I will post it!

    • one year ago
  37. experimentX
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    sure ... let me know too.

    • one year ago
  38. shivam_bhalla
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    @estudier ,Wolfram is getting this by Weierstrass substitution http://www.wolframalpha.com/input/?i=%E2%88%AB%282sqrt%282%29%29%281-t^2%29^2sqrt%28t%29%2F%281%2Bt^2%29^%285%2F2%29++dt @experimentX , check it please

    • one year ago
  39. experimentX
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    http://www.wolframalpha.com/input/?i=integrate+sqrt%28x++-+x^3%29+dx

    • one year ago
  40. experimentX
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    is it indefinite integral??

    • one year ago
  41. estudier
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    @shivam_bhalla Yes,seems like elliptic integral is the way...

    • one year ago
  42. estudier
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    Although I can't help thinking there ought to be a substitution that works Where is this question from?

    • one year ago
  43. RedPrince
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    this is my assignment and I post it!

    • one year ago
  44. shivam_bhalla
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    @RedPrince , I am sure there must be some typo in the question. BTW, in which grade are you studying in? (or What's your age?)

    • one year ago
  45. RedPrince
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    I am a student of BS Computer Science and I am 19 years old!

    • one year ago
  46. shivam_bhalla
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    Ohh. LOL I am just 17 and just completed my 12th grade

    • one year ago
  47. estudier
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    Maybe it needs two substitutions. I am surprised that they gave you such an integral.....

    • one year ago
  48. RedPrince
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    @estudier why you are surprised! I am a university student not a college student

    • one year ago
  49. shivam_bhalla
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    @estudier , I doubt so. I have tried various types of substitutions (till my level best) Until today, I never heard of elliptic integral

    • one year ago
  50. RedPrince
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    @shivam_bhalla me also first time I heard about elliptic integral

    • one year ago
  51. estudier
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    @satellite73 Anything spring to mind?

    • one year ago
  52. satellite73
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    lol university, not college. is the university not made up of colleges?

    • one year ago
  53. shivam_bhalla
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    LOL @satellite73

    • one year ago
  54. satellite73
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    no i am braid dead and i loathe techniques of integration. that is why they invented computers

    • one year ago
  55. estudier
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    lol

    • one year ago
  56. shivam_bhalla
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    Ok. @amistre64 should help when he comes online :)

    • one year ago
  57. shivam_bhalla
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    @amistre64 's profile picture itself is integration :P

    • one year ago
  58. estudier
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    (u^(1/2)(1-u^2)) du/(1-u^2)^(1/2) Best I can do, still gives elliptic

    • one year ago
  59. shivam_bhalla
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    @RedPrince , did you reconfirm the question ??

    • one year ago
  60. RedPrince
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    @shivam_bhalla Yes I asked to my teacher and he reconfirmed the question as: \[\int\limits \cos^3x \sqrt sinx dx\]

    • one year ago
  61. experimentX
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    is there limits?? or just indefinite integral??

    • one year ago
  62. experimentX
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    looks like this is quite easier that the original question http://www.wolframalpha.com/input/?i=integrate+cos^3x+sqrt%28sinx%29

    • one year ago
  63. shivam_bhalla
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    @RedPrince Yes, same question as @experimentX

    • one year ago
  64. experimentX
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    http://www.wolframalpha.com/input/?i=integrate+%281-x^2%29sqrt%28x%29 let sin(x) = u, cos^2x = 1 - u^2, cos(x)dx = du

    • one year ago
  65. shivam_bhalla
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    I think it would be better if t^2 = sinx 2tdt = cos x dx \[\int\limits_{}^{}(1-t^4)(t)(2t )(dt)\]

    • one year ago
  66. experimentX
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    well ... that solves our problem!!

    • one year ago
  67. shivam_bhalla
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    Good work @experimentX :) By the way, sometype wrong question teaches us new things :P :D

    • one year ago
  68. shivam_bhalla
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    *sometimes

    • one year ago
  69. experimentX
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    yea elliptical integrals!!

    • one year ago
  70. RedPrince
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    @shivam_bhalla yes sometimes they teaches us many things that are helpful in future....!! well thanks a lot all of you guys for helping me!!! espacially @experimentX ,@estudier and @shivam_bhalla

    • one year ago
  71. eliassaab
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    \[\frac{2}{5} \left(\sin ^{\frac{3}{2}}(x) \cos (x)-2 E\left(\left.\frac{1}{4} (\pi -2 x)\right|2\right)\right) \] This the answer I got from Mathematica. It does not seem to have an easy closed form

    • one year ago
  72. RedPrince
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    @eliassaab how this answer is come

    • one year ago
  73. eliassaab
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    This integral does not have a closed form in terms of usual functions, otherwise, mathematica would have found it.

    • one year ago
  74. mahmit2012
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    |dw:1336919117275:dw|

    • one year ago
  75. shivam_bhalla
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    @mahmit2012 , yes. @RedPrince said there was typo in original question he first mentioned

    • one year ago
  76. mahmit2012
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    |dw:1336919188247:dw|

    • one year ago
  77. RedPrince
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    @mahmit2012 yes you are true but how you get it!

    • one year ago
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