## RedPrince 3 years ago What is the integral of ∫(cos x)^2.sqrt{sin x}dx

1. lgbasallote

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

2. RedPrince

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

3. simran

|dw:1336216786369:dw|

4. simran

is dis a question?

5. estudier

Maybe it is doable with the Weierstrass substitution?

6. RedPrince

@simran yes it is aquestion

7. RedPrince

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

8. RedPrince

|dw:1336347046770:dw|

9. estudier

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

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

11. RedPrince

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

12. RedPrince

13. shivam_bhalla

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

14. RedPrince

@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

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

16. RedPrince

@shivam_bhalla I tried my best but failed!

17. shivam_bhalla

@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

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

19. RedPrince

@shivam_bhalla Ok w8 I try it

20. experimentX

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

21. shivam_bhalla

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

22. experimentX

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

23. RedPrince

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

24. shivam_bhalla

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

25. estudier

Did anybody try Weierstrass substitution?

26. RedPrince

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

27. RedPrince

I try Weierstrass substitution

28. shivam_bhalla

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

29. experimentX

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

30. shivam_bhalla

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

31. shivam_bhalla

@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

@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

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

34. experimentX

@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

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

36. RedPrince

@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

sure ... let me know too.

38. shivam_bhalla

@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
40. experimentX

is it indefinite integral??

41. estudier

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

42. estudier

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

43. RedPrince

this is my assignment and I post it!

44. shivam_bhalla

@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

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

46. shivam_bhalla

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

47. estudier

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

48. RedPrince

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

49. shivam_bhalla

@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

@shivam_bhalla me also first time I heard about elliptic integral

51. estudier

@satellite73 Anything spring to mind?

52. satellite73

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

53. shivam_bhalla

LOL @satellite73

54. satellite73

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

55. estudier

lol

56. shivam_bhalla

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

57. shivam_bhalla

@amistre64 's profile picture itself is integration :P

58. estudier

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

59. shivam_bhalla

@RedPrince , did you reconfirm the question ??

60. RedPrince

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

61. experimentX

is there limits?? or just indefinite integral??

62. experimentX

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

@RedPrince Yes, same question as @experimentX

64. experimentX

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

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

66. experimentX

well ... that solves our problem!!

67. shivam_bhalla

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

68. shivam_bhalla

*sometimes

69. experimentX

yea elliptical integrals!!

70. RedPrince

@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

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

72. RedPrince

@eliassaab how this answer is come

73. eliassaab

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

74. mahmit2012

|dw:1336919117275:dw|

75. shivam_bhalla

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

76. mahmit2012

|dw:1336919188247:dw|

77. RedPrince

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