Integral of $$$x^{4} e^{x}$$$

Find $$$\int x^{4} e^{x}\, dx$$$.

For the integral $$$\int{x^{4} e^{x} d x}$$$, use integration by parts $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$.

Let $$$\operatorname{u}=x^{4}$$$ and $$$\operatorname{dv}=e^{x} dx$$$.

Then $$$\operatorname{du}=\left(x^{4}\right)^{\prime }dx=4 x^{3} dx$$$ (steps can be seen ») and $$$\operatorname{v}=\int{e^{x} d x}=e^{x}$$$ (steps can be seen »).

So,

$${\color{red}{\int{x^{4} e^{x} d x}}}={\color{red}{\left(x^{4} \cdot e^{x}-\int{e^{x} \cdot 4 x^{3} d x}\right)}}={\color{red}{\left(x^{4} e^{x} - \int{4 x^{3} e^{x} d x}\right)}}$$

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=4$$$ and $$$f{\left(x \right)} = x^{3} e^{x}$$$:

$$x^{4} e^{x} - {\color{red}{\int{4 x^{3} e^{x} d x}}} = x^{4} e^{x} - {\color{red}{\left(4 \int{x^{3} e^{x} d x}\right)}}$$

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

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

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

The integral becomes

$$x^{4} e^{x} - 4 {\color{red}{\int{x^{3} e^{x} d x}}}=x^{4} e^{x} - 4 {\color{red}{\left(x^{3} \cdot e^{x}-\int{e^{x} \cdot 3 x^{2} d x}\right)}}=x^{4} e^{x} - 4 {\color{red}{\left(x^{3} e^{x} - \int{3 x^{2} e^{x} d x}\right)}}$$

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

$$x^{4} e^{x} - 4 x^{3} e^{x} + 4 {\color{red}{\int{3 x^{2} e^{x} d x}}} = x^{4} e^{x} - 4 x^{3} e^{x} + 4 {\color{red}{\left(3 \int{x^{2} e^{x} d x}\right)}}$$

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

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

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

The integral can be rewritten as

$$x^{4} e^{x} - 4 x^{3} e^{x} + 12 {\color{red}{\int{x^{2} e^{x} d x}}}=x^{4} e^{x} - 4 x^{3} e^{x} + 12 {\color{red}{\left(x^{2} \cdot e^{x}-\int{e^{x} \cdot 2 x d x}\right)}}=x^{4} e^{x} - 4 x^{3} e^{x} + 12 {\color{red}{\left(x^{2} e^{x} - \int{2 x e^{x} d x}\right)}}$$

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

$$x^{4} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 12 {\color{red}{\int{2 x e^{x} d x}}} = x^{4} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 12 {\color{red}{\left(2 \int{x e^{x} d x}\right)}}$$

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

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

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

Thus,

$$x^{4} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 {\color{red}{\int{x e^{x} d x}}}=x^{4} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 {\color{red}{\left(x \cdot e^{x}-\int{e^{x} \cdot 1 d x}\right)}}=x^{4} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 {\color{red}{\left(x e^{x} - \int{e^{x} d x}\right)}}$$

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

$$x^{4} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 x e^{x} + 24 {\color{red}{\int{e^{x} d x}}} = x^{4} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 x e^{x} + 24 {\color{red}{e^{x}}}$$

Therefore,

$$\int{x^{4} e^{x} d x} = x^{4} e^{x} - 4 x^{3} e^{x} + 12 x^{2} e^{x} - 24 x e^{x} + 24 e^{x}$$

Simplify:

$$\int{x^{4} e^{x} d x} = \left(x^{4} - 4 x^{3} + 12 x^{2} - 24 x + 24\right) e^{x}$$

Add the constant of integration:

$$\int{x^{4} e^{x} d x} = \left(x^{4} - 4 x^{3} + 12 x^{2} - 24 x + 24\right) e^{x}+C$$

$$$\int x^{4} e^{x}\, dx = \left(x^{4} - 4 x^{3} + 12 x^{2} - 24 x + 24\right) e^{x} + C$$$A