Question

Solve the integral: $\int x\sin xdx$

Short answer

The required answer is$-x\cos x+\sin x+c$.

Step-by-step solution

Step 1. Given information. 

We have given integral is$\int x\sin xdx$.

Step 2. Solve the integration by parts . 

We have,

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

and

localid="1648636397500" $$\begin{gathered} dv=\sin xdx \\ v=\int \sin xdx \\ v=-\cos x \end{gathered}$$

The formula of integration by parts is localid="1648645366461" $\int udv=uv-\int vdu$

$$\begin{gathered} \int x\sin xdx \\ =x(-\cos x)-\int (-\cos x)dx \\ =-x\cos x+\int \cos x \\ =-x\cos x+\sin x+c \end{gathered}$$