# Integral of $x \cos{\left(x \right)}$

The calculator will find the integral/antiderivative of $x \cos{\left(x \right)}$, with steps shown.

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Find $\int x \cos{\left(x \right)}\, dx$.

### Solution

For the integral $\int{x \cos{\left(x \right)} d x}$, use integration by parts $\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$.

Let $\operatorname{u}=x$ and $\operatorname{dv}=\cos{\left(x \right)} dx$.

Then $\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$ (steps can be seen here) and $\operatorname{v}=\int{\cos{\left(x \right)} d x}=\sin{\left(x \right)}$ (steps can be seen here).

So,

$${\color{red}{\int{x \cos{\left(x \right)} d x}}}={\color{red}{\left(x \cdot \sin{\left(x \right)}-\int{\sin{\left(x \right)} \cdot 1 d x}\right)}}={\color{red}{\left(x \sin{\left(x \right)} - \int{\sin{\left(x \right)} d x}\right)}}$$

The integral of the sine is $\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$:

$$x \sin{\left(x \right)} - {\color{red}{\int{\sin{\left(x \right)} d x}}} = x \sin{\left(x \right)} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

Therefore,

$$\int{x \cos{\left(x \right)} d x} = x \sin{\left(x \right)} + \cos{\left(x \right)}$$

$$\int{x \cos{\left(x \right)} d x} = x \sin{\left(x \right)} + \cos{\left(x \right)}+C$$
Answer: $\int{x \cos{\left(x \right)} d x}=x \sin{\left(x \right)} + \cos{\left(x \right)}+C$