# Need help with integral and change the variable

I have

$$\displaystyle \int_0^\pi \sqrt{x} ~\cos x ~dx$$

and I need to make change of the variable $u = \sin x$.

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Why in the world make that change of variables? Let $u=\sqrt x$ instead! – Ted Shifrin May 27 '13 at 17:27
The teacher want this, he told me first to make change on the limits of the integral -pi/2 0 pi/2 :( – user79727 May 27 '13 at 17:33
Well, that would mean $u=\pi/2-x$, but this won't get you anywhere. Perhaps your teacher wants you to think about symmetry, like $\displaystyle\int_{-\pi}^{\pi} x\cos x\,dx$, but that won't help here. – Ted Shifrin May 27 '13 at 17:37
This will not work. You'll end up with two integrals that can be done only by coming back to the original problem and doing it the right way!! Sorry to be a grouch. Sometimes — and I should know — teachers just make mistakes. – Ted Shifrin May 27 '13 at 17:52
Actually, your integrand doesn't have an elementary antiderivative; are you sure the problem wasn't $\int \sqrt{x} \cos^2 x dx$? – Zen May 27 '13 at 17:53

## 3 Answers

$\int_0^\pi\sqrt{x}\cos x~dx=\int_0^\pi x^{\frac{1}{2}}\sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{2n}}{(2n)!}dx=\int_0^\pi\sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{2n+\frac{1}{2}}}{(2n)!}dx=\left[\sum\limits_{n=0}^\infty\dfrac{(-1)^nx^{2n+\frac{3}{2}}}{(2n)!\left(2n+\dfrac{3}{2}\right)}\right]_0^\pi=\sum\limits_{n=0}^\infty\dfrac{2(-1)^n\pi^{2n+\frac{3}{2}}}{(2n)!(4n+3)}$

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Hint: Try this if you want: $$x\sqrt { x } =u\\ x={ u }^{ \frac { 2 }{ 3 } }\\ \sqrt { x } ={ u }^{ \frac { 1 }{ 3 } }\\ dx=\frac { 2 }{ 3 } { u }^{ \frac { -1 }{ 3 } }du\\ \int _{ 0 }^{ \pi } \sqrt { x } \cos x~ dx=\int _{ 0 }^{ \pi }{ u } ^{ \frac { 1 }{ 3 } }\cos { u } ^{ \frac { 2 }{ 3 } }\frac { 2 }{ 3 } { u }^{ \frac { -1 }{ 3 } }du=\frac { 2 }{ 3 } \int _{ 0 }^{ \pi\sqrt\pi }{ \cos { u } ^{ \frac { 2 }{ 3 } }du }$$

$\int { \cos { u } ^{ \frac { 2 }{ 3 } }du }$ does not have an analytical expression. You can try on wolframalpha.com .

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What earthly good is that?!! – Ted Shifrin May 27 '13 at 17:47

Write x as a function of u: you have $u=\sin(x)$ so $x=\arcsin(u)$. It gets a little sticky when you have to simplify $\cos(\arcsin(u))$; remember the identity $\sin^2\theta+\cos^2\theta=1$, letting $\theta=\arcsin u$ (arcsin gives you an angle, recall) you get $\cos(\arcsin(u))=\sqrt{1-u^2}$.

Actually, this integral doesn't have a closed form; are you sure the problem wasn't $\int \sqrt{x} \cos^2 x dx$

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This substitution isn't valid though, with $x\in [0,\pi]$, so we're going to have to break it up into two intervals and be very careful. – Ted Shifrin May 27 '13 at 17:43
Yes, i am sure it is the integral that i give :( – user79727 May 27 '13 at 17:53