Integration reduction formula

Question

How to create a reduction formula for the integral$$\int x \cos^n x \;\mathrm{d}x$$

I have tried everything i could think of, but I'm not even close to solving it. Really need help.

Answer 1

Hint:

A reduction formula for $$\int \cos^n x \;\mathrm{d}x$$

is $$\int \frac{\cos^{n-1}(x)\sin(x)}{n}$$

The additional 'x' term would be an application of integration by parts.

Answer 2

Let $I_n$ be the integral

$$I_n=\int x\cos^n(x)\,dx$$

Then, we have

$$\begin{align} I_{n+2}&=\int x\cos^2(x)\cos^n(x)\,dx\\\\ &=I_n-\int x\sin^2(x)\cos^n(x)\,dx\tag 1 \end{align}$$

Integrating by parts the integral on the right-hand side of $(1)$ with $u=x\sin(x)$ and $v=-\frac{\cos^{n+1}(x)}{n+1}$ yields

$$\begin{align} I_{n+2}&=I_n-\left(x\sin(x)\left(-\frac{\cos^{n+1}(x)}{n+1}\right)+\frac{1}{n+1}\int (x\cos(x)+\sin(x))\cos^{n+1}(x)\,dx\right)\\\\ &=I_n+\frac{x\sin(x)\cos^{n+1}(x)}{n+1}-\frac{1}{n+1}I_{n+2}+\frac{\cos^{n+2}(x)}{(n+1)(n+2)}\\\\ &=\frac{n+1}{n+2}I_n+\frac{x\sin(x)\cos^{n+1}(x)}{n+2}+\frac{\cos^{n+2}(x)}{(n+2)^2} \end{align}$$

Answer 3

You can actually evaluate this integral for general $n$ in terms of hypergeometric functions.