Question

Solve the integral:$\int x\cos \left(x\right)dx$.

Short answer

The required answer is$x\sin \left(x\right)+\cos \left(x\right)+c$.

Step-by-step solution

Step 1. Given information. 

We have given integral is$\int x\cos \left(x\right)dx$.

Step 2. Solve the integration by parts . 

We have, $u=x,du=dx$

and

$$\begin{gathered} dv=\cos \left(x\right)dx \\ v=\int \cos \left(x\right)dx \\ v=\sin x \end{gathered}$$

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

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