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

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

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

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

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

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

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

Thus,

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

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

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

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

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

Therefore,

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

Add the constant of integration:

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

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