Integral of $$$x^{3} e^{- 7 x}$$$

Find $$$\int x^{3} e^{- 7 x}\, dx$$$.

For the integral $$$\int{x^{3} e^{- 7 x} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=x^{3}$$$ and $$$\operatorname{dv}=e^{- 7 x} dx$$$.

Then $$$\operatorname{du}=\left(x^{3}\right)^{\prime }dx=3 x^{2} dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{e^{- 7 x} d x}=- \frac{e^{- 7 x}}{7}$$$ (steps can be seen »).

The integral can be rewritten as

$${\color{red}{\int{x^{3} e^{- 7 x} d x}}}={\color{red}{\left(x^{3} \cdot \left(- \frac{e^{- 7 x}}{7}\right)-\int{\left(- \frac{e^{- 7 x}}{7}\right) \cdot 3 x^{2} d x}\right)}}={\color{red}{\left(- \frac{x^{3} e^{- 7 x}}{7} - \int{\left(- \frac{3 x^{2} e^{- 7 x}}{7}\right)d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=- \frac{3}{7}$$$ and $$$f{\left(x \right)} = x^{2} e^{- 7 x}$$$:

$$- \frac{x^{3} e^{- 7 x}}{7} - {\color{red}{\int{\left(- \frac{3 x^{2} e^{- 7 x}}{7}\right)d x}}} = - \frac{x^{3} e^{- 7 x}}{7} - {\color{red}{\left(- \frac{3 \int{x^{2} e^{- 7 x} d x}}{7}\right)}}$$

For the integral $$$\int{x^{2} e^{- 7 x} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=x^{2}$$$ and $$$\operatorname{dv}=e^{- 7 x} dx$$$.

Then $$$\operatorname{du}=\left(x^{2}\right)^{\prime }dx=2 x dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{e^{- 7 x} d x}=- \frac{e^{- 7 x}}{7}$$$ (steps can be seen »).

The integral becomes

$$- \frac{x^{3} e^{- 7 x}}{7} + \frac{3 {\color{red}{\int{x^{2} e^{- 7 x} d x}}}}{7}=- \frac{x^{3} e^{- 7 x}}{7} + \frac{3 {\color{red}{\left(x^{2} \cdot \left(- \frac{e^{- 7 x}}{7}\right)-\int{\left(- \frac{e^{- 7 x}}{7}\right) \cdot 2 x d x}\right)}}}{7}=- \frac{x^{3} e^{- 7 x}}{7} + \frac{3 {\color{red}{\left(- \frac{x^{2} e^{- 7 x}}{7} - \int{\left(- \frac{2 x e^{- 7 x}}{7}\right)d x}\right)}}}{7}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=- \frac{2}{7}$$$ and $$$f{\left(x \right)} = x e^{- 7 x}$$$:

$$- \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} - \frac{3 {\color{red}{\int{\left(- \frac{2 x e^{- 7 x}}{7}\right)d x}}}}{7} = - \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} - \frac{3 {\color{red}{\left(- \frac{2 \int{x e^{- 7 x} d x}}{7}\right)}}}{7}$$

For the integral $$$\int{x e^{- 7 x} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=x$$$ and $$$\operatorname{dv}=e^{- 7 x} dx$$$.

Then $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{e^{- 7 x} d x}=- \frac{e^{- 7 x}}{7}$$$ (steps can be seen »).

Thus,

$$- \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} + \frac{6 {\color{red}{\int{x e^{- 7 x} d x}}}}{49}=- \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} + \frac{6 {\color{red}{\left(x \cdot \left(- \frac{e^{- 7 x}}{7}\right)-\int{\left(- \frac{e^{- 7 x}}{7}\right) \cdot 1 d x}\right)}}}{49}=- \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} + \frac{6 {\color{red}{\left(- \frac{x e^{- 7 x}}{7} - \int{\left(- \frac{e^{- 7 x}}{7}\right)d x}\right)}}}{49}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=- \frac{1}{7}$$$ and $$$f{\left(x \right)} = e^{- 7 x}$$$:

$$- \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} - \frac{6 x e^{- 7 x}}{343} - \frac{6 {\color{red}{\int{\left(- \frac{e^{- 7 x}}{7}\right)d x}}}}{49} = - \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} - \frac{6 x e^{- 7 x}}{343} - \frac{6 {\color{red}{\left(- \frac{\int{e^{- 7 x} d x}}{7}\right)}}}{49}$$

Let $$$u=- 7 x$$$.

Then $$$du=\left(- 7 x\right)^{\prime }dx = - 7 dx$$$ (steps can be seen »), and we have that $$$dx = - \frac{du}{7}$$$.

The integral can be rewritten as

$$- \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} - \frac{6 x e^{- 7 x}}{343} + \frac{6 {\color{red}{\int{e^{- 7 x} d x}}}}{343} = - \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} - \frac{6 x e^{- 7 x}}{343} + \frac{6 {\color{red}{\int{\left(- \frac{e^{u}}{7}\right)d u}}}}{343}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=- \frac{1}{7}$$$ and $$$f{\left(u \right)} = e^{u}$$$:

$$- \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} - \frac{6 x e^{- 7 x}}{343} + \frac{6 {\color{red}{\int{\left(- \frac{e^{u}}{7}\right)d u}}}}{343} = - \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} - \frac{6 x e^{- 7 x}}{343} + \frac{6 {\color{red}{\left(- \frac{\int{e^{u} d u}}{7}\right)}}}{343}$$

The integral of the exponential function is $$$\int{e^{u} d u} = e^{u}$$$:

$$- \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} - \frac{6 x e^{- 7 x}}{343} - \frac{6 {\color{red}{\int{e^{u} d u}}}}{2401} = - \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} - \frac{6 x e^{- 7 x}}{343} - \frac{6 {\color{red}{e^{u}}}}{2401}$$

Recall that $$$u=- 7 x$$$:

$$- \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} - \frac{6 x e^{- 7 x}}{343} - \frac{6 e^{{\color{red}{u}}}}{2401} = - \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} - \frac{6 x e^{- 7 x}}{343} - \frac{6 e^{{\color{red}{\left(- 7 x\right)}}}}{2401}$$

Therefore,

$$\int{x^{3} e^{- 7 x} d x} = - \frac{x^{3} e^{- 7 x}}{7} - \frac{3 x^{2} e^{- 7 x}}{49} - \frac{6 x e^{- 7 x}}{343} - \frac{6 e^{- 7 x}}{2401}$$

Simplify:

$$\int{x^{3} e^{- 7 x} d x} = \frac{\left(- 343 x^{3} - 147 x^{2} - 42 x - 6\right) e^{- 7 x}}{2401}$$

Add the constant of integration:

$$\int{x^{3} e^{- 7 x} d x} = \frac{\left(- 343 x^{3} - 147 x^{2} - 42 x - 6\right) e^{- 7 x}}{2401}+C$$

$$$\int x^{3} e^{- 7 x}\, dx = \frac{\left(- 343 x^{3} - 147 x^{2} - 42 x - 6\right) e^{- 7 x}}{2401} + C$$$A