Integral of $$$x e^{2} \cos{\left(5 x \right)}$$$

Find $$$\int x e^{2} \cos{\left(5 x \right)}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=e^{2}$$$ and $$$f{\left(x \right)} = x \cos{\left(5 x \right)}$$$:

$${\color{red}{\int{x e^{2} \cos{\left(5 x \right)} d x}}} = {\color{red}{e^{2} \int{x \cos{\left(5 x \right)} d x}}}$$

For the integral $$$\int{x \cos{\left(5 x \right)} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=x$$$ and $$$\operatorname{dv}=\cos{\left(5 x \right)} dx$$$.

Then $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{\cos{\left(5 x \right)} d x}=\frac{\sin{\left(5 x \right)}}{5}$$$ (steps can be seen »).

Thus,

$$e^{2} {\color{red}{\int{x \cos{\left(5 x \right)} d x}}}=e^{2} {\color{red}{\left(x \cdot \frac{\sin{\left(5 x \right)}}{5}-\int{\frac{\sin{\left(5 x \right)}}{5} \cdot 1 d x}\right)}}=e^{2} {\color{red}{\left(\frac{x \sin{\left(5 x \right)}}{5} - \int{\frac{\sin{\left(5 x \right)}}{5} d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{5}$$$ and $$$f{\left(x \right)} = \sin{\left(5 x \right)}$$$:

$$e^{2} \left(\frac{x \sin{\left(5 x \right)}}{5} - {\color{red}{\int{\frac{\sin{\left(5 x \right)}}{5} d x}}}\right) = e^{2} \left(\frac{x \sin{\left(5 x \right)}}{5} - {\color{red}{\left(\frac{\int{\sin{\left(5 x \right)} d x}}{5}\right)}}\right)$$

Let $$$u=5 x$$$.

Then $$$du=\left(5 x\right)^{\prime }dx = 5 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{5}$$$.

The integral can be rewritten as

$$e^{2} \left(\frac{x \sin{\left(5 x \right)}}{5} - \frac{{\color{red}{\int{\sin{\left(5 x \right)} d x}}}}{5}\right) = e^{2} \left(\frac{x \sin{\left(5 x \right)}}{5} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{5} d u}}}}{5}\right)$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{5}$$$ and $$$f{\left(u \right)} = \sin{\left(u \right)}$$$:

$$e^{2} \left(\frac{x \sin{\left(5 x \right)}}{5} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{5} d u}}}}{5}\right) = e^{2} \left(\frac{x \sin{\left(5 x \right)}}{5} - \frac{{\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{5}\right)}}}{5}\right)$$

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

$$e^{2} \left(\frac{x \sin{\left(5 x \right)}}{5} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{25}\right) = e^{2} \left(\frac{x \sin{\left(5 x \right)}}{5} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{25}\right)$$

Recall that $$$u=5 x$$$:

$$e^{2} \left(\frac{x \sin{\left(5 x \right)}}{5} + \frac{\cos{\left({\color{red}{u}} \right)}}{25}\right) = e^{2} \left(\frac{x \sin{\left(5 x \right)}}{5} + \frac{\cos{\left({\color{red}{\left(5 x\right)}} \right)}}{25}\right)$$

Therefore,

$$\int{x e^{2} \cos{\left(5 x \right)} d x} = \left(\frac{x \sin{\left(5 x \right)}}{5} + \frac{\cos{\left(5 x \right)}}{25}\right) e^{2}$$

Simplify:

$$\int{x e^{2} \cos{\left(5 x \right)} d x} = \frac{\left(5 x \sin{\left(5 x \right)} + \cos{\left(5 x \right)}\right) e^{2}}{25}$$

Add the constant of integration:

$$\int{x e^{2} \cos{\left(5 x \right)} d x} = \frac{\left(5 x \sin{\left(5 x \right)} + \cos{\left(5 x \right)}\right) e^{2}}{25}+C$$

$$$\int x e^{2} \cos{\left(5 x \right)}\, dx = \frac{\left(5 x \sin{\left(5 x \right)} + \cos{\left(5 x \right)}\right) e^{2}}{25} + C$$$A