Definite integral of $x\sin^n x$ from $0$ to $\pi/2$

Question

How to find \begin{equation*} \int_0^{\pi/2} x\sin^n x dx \end{equation*} where $n$ is a positive integer? I tried $y=x-\pi/4$ and that gives \begin{equation*} \frac{1}{2^{n/2}}\frac{\pi}{4}\int_{-\pi/4}^{\pi/4} (\sin y+\cos y)^n dy+\frac{1}{2^{n/2}}\int_{-\pi/4}^{\pi/4} y(\sin y+\cos y)^n dy. \end{equation*} Although some terms may be canceled, the summation of the others seems terrible.

Another method is integration by parts. Here I found the recurrence formula for $\sin^n x$: Integrating $\int \sin^n{x} \ dx$

So the first step is to have \begin{equation*} \left.\left(-\frac{1}{n}x\cos x\sin^{n-1} x+\frac{n-1}{n}x\int \sin^{n-2} xdx\right)\right|_0^{\pi/2}-\int_0^{\pi/2}\left(-\frac{1}{n}\cos x\sin^{n-1} x+\frac{n-1}{n}\int_0^x \sin^{n-2} t dt\right)dx \end{equation*} the evaluations at $0$ and $\pi/2$ and the integral of $\cos x\sin^{n-1}x$ seem OK, but what about the second term? Any better method for this problem?

Answer 1

Do the same thing: let the integral be $I_n$, $$ \begin{align*} &\phantom{=}\int_0^{\pi/2} x \sin(x)^n \mathrm dx = -\int_0^{\pi/2} x \sin(x)^{n-1}\mathrm d(\cos(x)) \\ &=\left. x \sin(x)^{n-1} \cos(x) \right|_{\pi/2}^0 - \int_{\pi/2}^0 \cos(x) (\sin(x)^{n-1} + (n-1)x \sin(x)^{n-2} \cos(x))\mathrm d x \\ &= 0 + \int_0^{\pi/2} \sin(x)^{n-1}\mathrm d (\sin(x)) +(n-1) \int_0^{\pi/2} x\sin(x)^{n-2}(1 - \sin(x)^2)\mathrm dx\\ &= \frac 1n + (n-1) I_{n-2} - (n-1)I_n , \end{align*} $$ i.e. $$ \fbox{$nI_n =\dfrac 1n + (n-1)I_{n-2}, $} $$ with $$ I_1 = 1,\quad I_2 = \frac 14 + \frac {\pi^2} {16}, $$ since \begin{align*} I_2&=\int_0^{\pi/2} x\sin(x)^2\mathrm dx \\ &= \left. x \sin(x) \cos(x)\right|_{\pi/2}^0 + \int_0^{\pi/2} \cos(x) (\sin(x)+ x\cos(x)) \mathrm dx \\ &= 0 + \frac 12 + \int_0^{\pi/2} x(1 - \sin(x)^2) \mathrm dx\\ &= \frac 12 + \frac {\pi^2} 8 - I_2\\ &\implies I_2 = \frac 14 + \frac {\pi^2}{16}. \end{align*} To go further, we are going to solve the recursive relation, but seems gruesome. So I let you take it from here.

UPDATE

According to wiki, we have a systematic method to solve this. First consider the terms with even index,

Let $J_{n} = I_{2n}$, then $$ J_n = \frac 1{4n^2} + \frac {2n-1}{2n}J_{n-1}, $$ then divide both side of the equation by $(2n-1)!!/(2n)!!$: $$ \frac {J_n}{\dfrac {(2n-1)!!}{(2n)!!}} = \frac 1 {4n^2} \frac {(2n)!!}{(2n-1)!!}+ \frac {J_{n-1}}{\dfrac {(2n-3)!!}{(2n-2)!!}}, $$ now let $P_n = J_n (2n)!!/(2n-1)!!$, then $$ P_n = \frac 1{2n} \frac {(2n-2)!!}{(2n-1)!!} + P_{n-1}, $$ then $$ P_n = \sum_2^n \frac 1{2k}\frac {(2k-2)!!}{(2k-1)!!} + P_1, $$ which is pretty much for deriving a concrete number of $I_n$.

Answer 2

Assuming that $n$ is odd ($n=2m+1$), we may consider the triangle wave $$ t(x) = \frac{4}{\pi}\sum_{n\geq 0}\frac{(-1)^n}{(2n+1)^2}\,\sin((2n+1)x)$$ and the fact that, by symmetry, $$ \int_{0}^{\pi/2}x\sin^n(x)\,dx = \frac{1}{4}\int_{0}^{2\pi}t(x)\sin^n(x)\,dx. $$ On the other hand the Fourier sine series of $\sin^n(x)$ is straightforward to find: $$ \sin^n(x) = \frac{1}{2^{n-1}}\sum_{k=0}^{m}\binom{n}{m-k} (-1)^k \sin((2k+1)x)$$ hence by the orthogonality relations: $$\boxed{ \int_{0}^{\pi/2}x \sin^{2m+1}(x)\,dx = \frac{1}{2^{2m+1}}\sum_{k=0}^{m}\binom{2m+1}{m-k}\frac{1}{(2k+1)^2}}\tag{odd}$$ If $n$ is even we may similarly consider the Fourier cosine series of $\sin^n(x)$ and replace $x$ with the $\pi$-periodic continuation of $\frac{\pi}{2}-\left|\frac{\pi}{2}-x\right|$ over $[0,\pi]$, i.e. just a different triangle wave:

$$ T(x) = \frac{\pi}{4}-\frac{2}{\pi}\sum_{n\geq 0}\frac{\cos((4n+2)x)}{(2n+1)^2} = |t(x)|$$ leading to: $$ \boxed{\int_{0}^{\pi/2}x \sin^{2m}(x)\,dx = \frac{\binom{2m}{m}\pi^2}{8\cdot 4^m}+\frac{1}{2^{2m}}\sum_{\substack{1\leq k\leq m\\k\text{ odd}}}\binom{2m}{m-k}\frac{1}{k^2}}\tag{even}$$ We may notice that the substitution $x\mapsto \arcsin x$, together with the Maclaurin series of $\arcsin^2$, provides a further hypergeometric representation for the given integral:

$$\begin{eqnarray*} \int_{0}^{\pi/2}x\sin^n(x)\,dx = \int_{0}^{1}\frac{\arcsin x}{\sqrt{1-x^2}}x^n\,dx &=&\sum_{m\geq 1}\frac{4^m}{2m(2m+n)\binom{2m}{m}}\\&=&\frac{1}{n+2}\cdot\phantom{}_3 F_2\left(1,1,1+\tfrac{n}{2};\tfrac{3}{2},2+\tfrac{n}{2};1\right).\end{eqnarray*}$$ The equality between the RHS of the last line and $(\text{odd})$ or $(\text{even})$ is probably a consequence of general transformation identities for hypergeometric $\phantom{}_3 F_2(\ldots;1)$ functions.

Answer 3

$I_0=\frac{\pi^2}8$ and $I_1=1$ and for $n\ge2$, $$ \begin{align} I_n &=\int_0^{\pi/2}x\sin^n(x)\,\mathrm{d}x\tag1\\ &=-\int_0^{\pi/2}x\sin^{n-1}(x)\,\mathrm{d}\cos(x)\tag2\\ &=(n-1)\int_0^{\pi/2}x\cos^2(x)\sin^{n-2}(x)\,\mathrm{d}x+\int_0^{\pi/2}\cos(x)\sin^{n-1}(x)\,\mathrm{d}x\tag3\\ &=(n-1)\int_0^{\pi/2}x\sin^{n-2}(x)\,\mathrm{d}x-(n-1)I_n+\frac1n\tag4\\ &=\frac{n-1}nI_{n-2}+\frac1{n^2}\tag5\\ \end{align} $$ Explanation:
$(2)$: prepare to integrate by parts
$(3)$: integrate by parts
$(4)$: $\cos^2(x)=1-\sin^2(x)$ and the $\sin^2(x)$ part replicates $I_n$
$(5)$: add $(n-1)I_n$ to both sides and divide by $n$

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Using $I_0$ and $(5)$, we can compute $I_{2n}$: $$ \begin{align} I_{2n} &=\frac{(2n-1)!!}{(2n)!!}\left[\frac{\pi^2}8+\sum_{k=0}^{n-1}\frac{(2n-2k-2)!!}{(2n-2k-1)!!}\frac1{2n-2k}\right]\\ &=\frac{\binom{2n}{n}}{4^n}\left[\frac{\pi^2}8+\sum_{k=0}^{n-1}\frac{4^{n-k}}{\binom{2n-2k}{n-k}}\frac1{(2n-2k)^2}\right] \end{align} $$

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Using $I_1$ and $(5)$, we can compute $I_{2n+1}$: $$ \begin{align} I_{2n+1} &=\frac{(2n)!!}{(2n+1)!!}\left[1+\sum_{k=0}^{n-1}\frac{(2n-2k-1)!!}{(2n-2k)!!}\frac1{2n-2k+1}\right]\\ &=\frac1{2n+1}\frac{4^n}{\binom{2n}{n}}\left[1+\sum_{k=0}^{n-1}\frac{\binom{2n-2k}{n-k}}{4^{n-k}}\frac1{2n-2k+1}\right] \end{align} $$