# How do you integrate x * cos^2 (x)?

Nov 7, 2016

${x}^{2} / 4 + \left(\frac{x}{4}\right) \sin 2 x + \cos \frac{2 x}{8} + c$.where c is integration constant .

#### Explanation:

$\int \left(x {\cos}^{2} x\right) \mathrm{dx}$
$= \left(\frac{1}{2}\right) \int \left(x 2 {\cos}^{2} x\right) \mathrm{dx}$
now $\int \left(x 2 {\cos}^{2} x\right) \mathrm{dx}$
=$\int \left\{x \left(1 + \cos 2 x\right)\right\} \mathrm{dx}$
=$\int x \mathrm{dx} + \int \left(x \cos 2 x\right) \mathrm{dx}$
[integration by parts]
=${x}^{2} / 2 + x \left(\int \left(\cos 2 x\right) \mathrm{dx}\right) - \int \left[\left(\frac{\mathrm{dx}}{\mathrm{dx}}\right) \int \left(\cos 2 x\right) \mathrm{dx}\right] \mathrm{dx}$
=${x}^{2} / 2 + \frac{x \sin \left(2 x\right)}{2}$$- \int \left(\frac{\sin 2 x}{2}\right) \mathrm{dx}$
=${x}^{2} / 2 + \frac{x \sin \left(2 x\right)}{2}$$+ \frac{\cos 2 x}{4}$$+ c$
hence the value of the integral :${x}^{2} / 4 + \frac{x \sin \left(2 x\right)}{4} + \cos \frac{2 x}{8} + c$,where c is constant of integration.