How to effectively calculate $$\int_0^\pi \sin^3x \sin(nx)dx$$
I think by the formula: $$\sin 3x=3\sin x -4\sin^3x $$ So we get $$\sin^3x = \frac{3\sin x-\sin 3x}{4}$$
Plug in we obtain $$\frac{1}{4}\int_0^\pi (3\sin x-\sin 3x) \sin(nx)dx$$
Then how do I proceed?
On
You're doing fine. To finish, you can use the following Hint : $\sin(p)\sin(q)=\frac{\cos(p-q)-\cos(p+q)}{2}$.
On
HINT:
Through integration by parts, the integral $\sf{\int\sin mx\sin nx\,dx}$ is equivalent to $$\sf{-\frac1n\sin mx\cos nx+\frac mn\int \cos mx\cos nx\,dx}$$ and use integration by parts again for $\sf{\int\cos mx\cos nx\,dx}$ to get $$\sf{\frac1n\cos mx\sin nx-\frac mn\int\sin mx\sin nx\,dx}$$ and equate the two common integrals to get that $$\sf{\int\sin mx\sin nx\,dx=-\frac1n\sin mx\cos nx+\frac m{n^2}\cos mx\sin nx-\frac{m^2}{n^2}\int\sin mx\sin nx\,dx.}$$
On
We proceed exactly as how you started, and then use the product-to-sum formula: $\sin(a) \sin(b) = \frac{1}{2} (\cos(a-b) - \cos(a+b))$. If $n \neq 1, 3$:
$$\begin{aligned} I_n &= \displaystyle \int_0^{\pi} \sin^3(x) \sin(nx) \; \mathrm{d}x\\ &= \frac{1}{4} \displaystyle \int_0^{\pi} (3 \sin(x) - \sin(3x)) \sin(nx) \; \mathrm{d}x\\ &= \frac{1}{4} \displaystyle \int_0^{\pi} 3 \sin(nx) \sin(x) - \sin(nx) \sin(3x) \; \mathrm{d}x\\ &= \frac{1}{4} \displaystyle \int_0^{\pi} \frac{3}{2} \left ( \cos \left ((n-1) x \right ) - \cos \left ( (n+1) x \right ) \right ) - \frac{1}{2} \left ( \cos \left ( (n-3) x \right ) - \cos \left ( (n+3) x \right ) \right ) \; \mathrm{d}x\\ &= \frac{1}{8} \left [ \frac{3}{n-1} \sin \left ( (n-1) x \right ) - \frac{3}{n+1} \sin \left ( (n+1) x \right ) - \frac{1}{n-3} \sin \left ( (n-3) x \right ) + \frac{1}{n+3} \sin \left ( (n+3) x \right ) \right ]_0^{\pi}\\ &= 0 \end{aligned}$$
Thus, if $n \neq 1, 3$, then the integral evaluates to $0$. If $n=1$, we have:
$$\begin{aligned} I_1 &= \frac{1}{4} \displaystyle \int_0^{\pi} \frac{3}{2} \left ( \cos \left ((n-1) x \right ) - \cos \left ( (n+1) x \right ) \right ) - \frac{1}{2} \left ( \cos \left ( (n-3) x \right ) - \cos \left ( (n+3) x \right ) \right ) \; \mathrm{d}x\\ &= \frac{1}{8} \displaystyle \int_0^{\pi} 3 - 3\cos(2x)- \cos(-2x) + \cos(4x) \; \mathrm{d}x\\ &= \frac{1}{8} \left [3x - 3 \sin(2x) + \frac{3}{4} \sin(4x) \right ]_0^{\pi}\\ &= \frac{3\pi}{8} \end{aligned}$$
If $n=3$, we have:
$$\begin{aligned} I_3 &= \frac{1}{4} \displaystyle \int_0^{\pi} \frac{3}{2} \left ( \cos \left ((n-1) x \right ) - \cos \left ( (n+1) x \right ) \right ) - \frac{1}{2} \left ( \cos \left ( (n-3) x \right ) - \cos \left ( (n+3) x \right ) \right ) \; \mathrm{d}x\\ &= \frac{1}{8} \displaystyle \int_0^{\pi} 3\cos(2x)-3\cos(4x)-1+\cos(6x) \; \mathrm{d}x\\ &= \frac{1}{8} \left [\frac{3}{2} \sin(2x) - \frac{3}{4}\sin(4x) - x + \frac{1}{6} \sin(6x) \right ]_0^{\pi}\\ &= -\frac{\pi}{8} \end{aligned}$$
Thus, over all cases we have $$I_n = \begin{cases} 0 \quad &\text{if } n \neq 1, 3\\ \frac{3\pi}{8} \quad &\text{if } n = 1\\ -\frac{\pi}{8} \quad &\text{if } n = 3 \end{cases}$$
On
$$J=\int_0^\pi \sin^3x \sin(nx)dx$$
Using Werner Formulas , $$8J=3(I_{n-1}-I_{n+1})-(I_{n-3}-I_{n+3})$$
$$\text{ where } I_m=\int_0^\pi\cos mx\ dx=\begin{cases}\pi &\mbox{if } m=0 \\ 0 & \mbox{other integer values of }m \end{cases} $$
Set $n=1,-1,-3,3$
Maybe it's a more advanced tool. I'm expose anyway my solution (for people who are interested). Remark that $x\mapsto \sin(x)\sin(nx)$ is even for all $n\in\mathbb N^*$, therefore, $$\int_0^{\pi}\sin^3(x)\sin(nx)\,\mathrm d x=\frac{1}{2}\int_{-\pi}^\pi\sin^3(x)\sin(nx)\,\mathrm d x.$$
As the OP remarked, $$\sin^3(x)=\frac{3}{4}\sin(x)-\frac{1}{4}\sin(3x).$$ Therefore, using Fourier series, we immediately obtain
$$\int_{-\pi}^\pi\sin^3(x)\sin(nx)\,\mathrm d x=\begin{cases}\frac{3\pi}{4}&n=1\\ -\frac{\pi}{4}&n=3\\ 0&\text{otherwise}\end{cases}.$$