Quantcast

A community for students.

Here's the question you clicked on:

55 members online
  • 0 replying
  • 0 viewing

RedPrince

  • 3 years ago

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

  • This Question is Closed
  1. lgbasallote
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  2. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @lgbasallote It is not working. I already checked it.....!!

  3. simran
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1336216786369:dw|

  4. simran
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    is dis a question?

  5. estudier
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Maybe it is doable with the Weierstrass substitution?

  6. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @simran yes it is aquestion

  7. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @estudier what you want to say.... explain please

  8. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1336347046770:dw|

  9. estudier
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Weierstrass sub sin x = 2t/(1+t^2) cos x = (1-t^2)/(1+t^2) dx = 2 dt/(1+t^2)

  10. badi
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    \[2(7\sin ^{3/2}x-3\sin ^{7/2})/21\]

  11. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  12. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @badi explain please

  13. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    One tip @RedPrince To write as fractions in latex Use the folowing syntax frac{numerator}{denominator}

  14. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @shivam_bhalla sorry I am a new user on this site! I will keep it in my mind in future. thanks for tip

  15. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    No problem. :). trying to solve your problem :D

  16. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @shivam_bhalla I tried my best but failed!

  17. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @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

  18. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    Oops in part 2 of the problem, it should be \[-\int\limits\limits_{}^{}{(2p^2-1)}dp\]

  19. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @shivam_bhalla Ok w8 I try it

  20. experimentX
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  21. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @experimentX , I am getting the answer with my method :D

  22. experimentX
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  23. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  24. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    Yes. Sorry for the mistake in part 2 of the problem. let me correct it and tell you

  25. estudier
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Did anybody try Weierstrass substitution?

  26. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @shivam_bhalla it's ok!!! I am waiting and try Weierstrass substitution as @estudier is said

  27. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I try Weierstrass substitution

  28. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @estudier , I tried but getting some unsolvable integral. why don't you try and see :D

  29. experimentX
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  30. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @experimentX , that is the solution for part 1. Now I am stuck at part 2 of the integration

  31. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @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

  32. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @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

  33. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @experimentX ,Looks like we need to learn elliptical integral.

  34. experimentX
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @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

  35. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    LOL. looks like I really need to sleep now. Sorry @RedPrince .@experimentX .

  36. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @shivam_bhalla it's ok dear text my friend and he reply that he solve it! when I get answer I will post it!

  37. experimentX
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    sure ... let me know too.

  38. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @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

  39. experimentX
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    http://www.wolframalpha.com/input/?i=integrate+sqrt%28x++-+x^3%29+dx

  40. experimentX
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    is it indefinite integral??

  41. estudier
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @shivam_bhalla Yes,seems like elliptic integral is the way...

  42. estudier
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Although I can't help thinking there ought to be a substitution that works Where is this question from?

  43. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    this is my assignment and I post it!

  44. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @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?)

  45. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    I am a student of BS Computer Science and I am 19 years old!

  46. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    Ohh. LOL I am just 17 and just completed my 12th grade

  47. estudier
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    Maybe it needs two substitutions. I am surprised that they gave you such an integral.....

  48. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @estudier why you are surprised! I am a university student not a college student

  49. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

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

  50. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @shivam_bhalla me also first time I heard about elliptic integral

  51. estudier
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    @satellite73 Anything spring to mind?

  52. satellite73
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    lol university, not college. is the university not made up of colleges?

  53. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    LOL @satellite73

  54. satellite73
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    no i am braid dead and i loathe techniques of integration. that is why they invented computers

  55. estudier
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    lol

  56. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    Ok. @amistre64 should help when he comes online :)

  57. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @amistre64 's profile picture itself is integration :P

  58. estudier
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    (u^(1/2)(1-u^2)) du/(1-u^2)^(1/2) Best I can do, still gives elliptic

  59. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @RedPrince , did you reconfirm the question ??

  60. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @shivam_bhalla Yes I asked to my teacher and he reconfirmed the question as: \[\int\limits \cos^3x \sqrt sinx dx\]

  61. experimentX
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    is there limits?? or just indefinite integral??

  62. experimentX
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

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

  63. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @RedPrince Yes, same question as @experimentX

  64. experimentX
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    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

  65. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

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

  66. experimentX
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    well ... that solves our problem!!

  67. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    Good work @experimentX :) By the way, sometype wrong question teaches us new things :P :D

  68. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    *sometimes

  69. experimentX
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 1

    yea elliptical integrals!!

  70. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @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

  71. eliassaab
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

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

  72. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @eliassaab how this answer is come

  73. eliassaab
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    This integral does not have a closed form in terms of usual functions, otherwise, mathematica would have found it.

  74. mahmit2012
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1336919117275:dw|

  75. shivam_bhalla
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 3

    @mahmit2012 , yes. @RedPrince said there was typo in original question he first mentioned

  76. mahmit2012
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    |dw:1336919188247:dw|

  77. RedPrince
    • 3 years ago
    Best Response
    You've already chosen the best response.
    Medals 0

    @mahmit2012 yes you are true but how you get it!

  78. Not the answer you are looking for?
    Search for more explanations.

    • Attachments:

Ask your own question

Sign Up
Find more explanations on OpenStudy
Privacy Policy

Your question is ready. Sign up for free to start getting answers.

spraguer (Moderator)
5 → View Detailed Profile

is replying to Can someone tell me what button the professor is hitting...

23

  • Teamwork 19 Teammate
  • Problem Solving 19 Hero
  • You have blocked this person.
  • ✔ You're a fan Checking fan status...

Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.

This is the testimonial you wrote.
You haven't written a testimonial for Owlfred.