Integral of $$$x \cos{\left(\pi n x \right)}$$$ with respect to $$$x$$$

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

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

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

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

The integral can be rewritten as

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{\pi n}$$$ and $$$f{\left(x \right)} = \sin{\left(\pi n x \right)}$$$:

$$- {\color{red}{\int{\frac{\sin{\left(\pi n x \right)}}{\pi n} d x}}} + \frac{x \sin{\left(\pi n x \right)}}{\pi n} = - {\color{red}{\frac{\int{\sin{\left(\pi n x \right)} d x}}{\pi n}}} + \frac{x \sin{\left(\pi n x \right)}}{\pi n}$$

Let $$$u=\pi n x$$$.

Then $$$du=\left(\pi n x\right)^{\prime }dx = \pi n dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{\pi n}$$$.

Therefore,

$$\frac{x \sin{\left(\pi n x \right)}}{\pi n} - \frac{{\color{red}{\int{\sin{\left(\pi n x \right)} d x}}}}{\pi n} = \frac{x \sin{\left(\pi n x \right)}}{\pi n} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{\pi n} d u}}}}{\pi n}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{\pi n}$$$ and $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$\frac{x \sin{\left(\pi n x \right)}}{\pi n} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{\pi n} d u}}}}{\pi n} = \frac{x \sin{\left(\pi n x \right)}}{\pi n} - \frac{{\color{red}{\frac{\int{\sin{\left(u \right)} d u}}{\pi n}}}}{\pi n}$$

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

$$\frac{x \sin{\left(\pi n x \right)}}{\pi n} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{\pi^{2} n^{2}} = \frac{x \sin{\left(\pi n x \right)}}{\pi n} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{\pi^{2} n^{2}}$$

Recall that $$$u=\pi n x$$$:

$$\frac{x \sin{\left(\pi n x \right)}}{\pi n} + \frac{\cos{\left({\color{red}{u}} \right)}}{\pi^{2} n^{2}} = \frac{x \sin{\left(\pi n x \right)}}{\pi n} + \frac{\cos{\left({\color{red}{\pi n x}} \right)}}{\pi^{2} n^{2}}$$

Therefore,

$$\int{x \cos{\left(\pi n x \right)} d x} = \frac{x \sin{\left(\pi n x \right)}}{\pi n} + \frac{\cos{\left(\pi n x \right)}}{\pi^{2} n^{2}}$$

Simplify:

$$\int{x \cos{\left(\pi n x \right)} d x} = \frac{\pi n x \sin{\left(\pi n x \right)} + \cos{\left(\pi n x \right)}}{\pi^{2} n^{2}}$$

Add the constant of integration:

$$\int{x \cos{\left(\pi n x \right)} d x} = \frac{\pi n x \sin{\left(\pi n x \right)} + \cos{\left(\pi n x \right)}}{\pi^{2} n^{2}}+C$$

$$$\int x \cos{\left(\pi n x \right)}\, dx = \frac{\pi n x \sin{\left(\pi n x \right)} + \cos{\left(\pi n x \right)}}{\pi^{2} n^{2}} + C$$$A