Question

Integrate with respect to x:
(i) $\frac{x}{nx}$
(ii) $x{\sin }^{2}x$

Answer 1

$\left(i\right)\int \frac{x}{nx}dx=\int \frac{1}{n}dx=\frac{x}{n}+C\left(ii\right)\int x{\sin }^{2}xdx=\int x.\frac{1}{2}(1-\cos 2x)dx=\frac{1}{2}\int xdx-\frac{1}{2}\int x\cos 2xdx=\frac{x^2}{4}-\frac{1}{2}[x\int \cos 2xdx-\int [\frac{dx}{dx}.\int \cos 2xdx]dx]\left(integratingbypart\right)=\frac{x^2}{4}-\frac{x\sin 2x}{4}-\frac{\cos 2x}{4}+C=\frac{x^2-x\sin 2x-\cos 2x}{4}+C$