Integral/Powers of sine/Recursion/Example

A primitive function for the sine function $\sin x$ is $-\cos x$. In order to find a primitive function for ${\sin }^{n}x$, we use integration by parts to get a recursive relation to a power with a smaller exponent. To make this more precise, we work over an interval, the primitive function shall start at $0$ and have the value $0$ there. For $n\ge 2$, with integration by parts, we get

$\begin{array}{cc}{\int }_0^x{\sin }^{n}tdt& ={\int }_0^x{\sin }^{n-2}t\cdot {\sin }^{2}tdt\\ & ={\int }_0^x{\sin }^{n-2}t\cdot (1-{\cos }^{2}t)dt\\ & ={\int }_0^x{\sin }^{n-2}tdt-{\int }_0^x({\sin }^{n-2}t\cos t)\cos tdt\\ & ={\int }_0^x{\sin }^{n-2}tdt-\frac{{\sin }^{n-1}t}{n-1}\cos t{|}_0^x-\frac{1}{n-1}\left({\int }_0^x{\sin }^{n}tdt\right).\end{array}$

Multiplication with $n-1$ and rearranging yields

$n{\int }_0^x{\sin }^{n}tdt=(n-1){\int }_0^x{\sin }^{n-2}tdt-{\sin }^{n-1}x\cos x.$

In particular, for $n=2$, we have

${\int }_0^x{\sin }^{2}tdt=\frac{1}{2}(x-\sin x\cos x).$