Evaluate the following integrals:

Let $\mathrm{I}=\int \mathrm{x} \cos \mathrm{x} \mathrm{dx}$

We know that, $\int \mathrm{UV}=\mathrm{U} \int \mathrm{V} \mathrm{dv}-\int \frac{\mathrm{d}}{\mathrm{dx}} \mathrm{U} \int \mathrm{V} \mathrm{dv}$

Using integration by parts,

$I=x \int \cos x d x-\int \frac{d}{d x} x \int \cos x d x I=\int x \cos x d x$

We have, $\int \sin x=-\cos x, \int \cos x=\sin x$

$=x \times \sin x-\int \sin x d x$

$=x \sin x+\cos x+c$