Question

Solve the integral: $\int x\sin {\left(x\right)}^{2}dx$.

Short answer

The required answer is$-\frac{\cos {\left(x\right)}^2}{2}+c$.

Step-by-step solution

Step 1. Given information. 

We have given integral is$\int x\sin {\left(x\right)}^{2}dx$.

Step 2. Solve the integration by parts . 

We have, $$\begin{gathered} u=x \\ du=dx \end{gathered}$$

and

$$\begin{gathered} dv={\sin }^{2}xdx \\ v=\int {\sin }^{2}xdx \\ v=\frac{x}{2}-\frac{\sin 2x}{4} \end{gathered}$$

The formula of integration by parts is $\int udv=uv-\int vdu$

$$\begin{gathered} \int x\sin {\left(x\right)}^{2}dx \\ =x\left[\frac{x}{2}-\frac{\sin 2x}{4}\right]-\int \frac{x}{2}-\frac{\sin 2x}{4}dx \\ =\frac{x^2}{2}-\frac{x\sin 2x}{4}-\frac{x^2}{4}+\int \frac{\sin 2x}{4}dx \\ =-\frac{\cos x^2}{2}+c \end{gathered}$$