Question

Solve: $\underset{0}{\overset{\pi /4}{\int }}\left(x\cos x-\cos 3x\right)dx$

Answer 1

$\begin{array}{c}\underset{0}{\overset{\pi /4}{\int }}\left(x\cos x-\cos 3x\right)dx\\ =\underset{0}{\overset{\pi /4}{\int }}x\cos xdx-\underset{0}{\overset{\pi /4}{\int }}\cos 3xdx\\ ={\left[x\int \cos xdx-\int \sin xdx\right]}_0^{\pi /4}-{\left[\frac{\sin 3x}{x}\right]}_0^{\pi /4}\\ ={\left[x\sin x-\cos x\right]}_0^{\pi /4}-\left[\frac{\sin 3\pi /4}{3}\right]\\ =\left[\frac{\pi }{4}.\frac{1}{2}+\frac{1}{2}-1\right]-\left[\frac{1/\sqrt{2}}{3}\right]\\ =\frac{\pi }{4\sqrt{2}}-1-\frac{\sqrt{2}}{3}\end{array}$