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

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- RedPrince

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

- jamiebookeater

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- lgbasallote

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

- RedPrince

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

- anonymous

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## More answers

- anonymous

is dis a question?

- anonymous

Maybe it is doable with the Weierstrass substitution?

- RedPrince

@simran yes it is aquestion

- RedPrince

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

- RedPrince

|dw:1336347046770:dw|

- anonymous

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

- anonymous

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

- 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!

- RedPrince

@badi explain please

- anonymous

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

- RedPrince

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

- anonymous

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

- RedPrince

@shivam_bhalla I tried my best but failed!

- anonymous

@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

- anonymous

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

- RedPrince

@shivam_bhalla Ok w8 I try it

- experimentX

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

- anonymous

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

- experimentX

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

- RedPrince

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

- anonymous

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

- anonymous

Did anybody try Weierstrass substitution?

- RedPrince

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

- RedPrince

I try Weierstrass substitution

- anonymous

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

- experimentX

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

- anonymous

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

- anonymous

@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

- 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

- anonymous

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

- 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

- anonymous

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

- 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!

- experimentX

sure ... let me know too.

- anonymous

@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

- experimentX

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

- experimentX

is it indefinite integral??

- anonymous

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

- anonymous

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

- RedPrince

this is my assignment and I post it!

- anonymous

@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?)

- RedPrince

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

- anonymous

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

- anonymous

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

- RedPrince

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

- anonymous

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

- RedPrince

@shivam_bhalla me also first time I heard about elliptic integral

- anonymous

@satellite73 Anything spring to mind?

- anonymous

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

- anonymous

LOL @satellite73

- anonymous

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

- anonymous

lol

- anonymous

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

- anonymous

@amistre64 's profile picture itself is integration :P

- anonymous

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

- anonymous

@RedPrince , did you reconfirm the question ??

- RedPrince

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

- experimentX

is there limits?? or just indefinite integral??

- experimentX

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

- anonymous

@RedPrince Yes, same question as @experimentX

- 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

- anonymous

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

- experimentX

well ... that solves our problem!!

- anonymous

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

- anonymous

*sometimes

- experimentX

yea elliptical integrals!!

- 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

- anonymous

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

- RedPrince

@eliassaab how this answer is come

- anonymous

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

- anonymous

|dw:1336919117275:dw|

- anonymous

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

- anonymous

|dw:1336919188247:dw|

- RedPrince

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

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