# Integral of $x e^{x^{2}}$

The calculator will find the integral/antiderivative of $x e^{x^{2}}$, with steps shown.

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Find $\int x e^{x^{2}}\, dx$.

### Solution

Let $u=x^{2}$.

Then $du=\left(x^{2}\right)^{\prime }dx = 2 x dx$ (steps can be seen here), and we have that $x dx = \frac{du}{2}$.

The integral becomes

$$\color{red}{\int{x e^{x^{2}} d x}} = \color{red}{\int{\frac{e^{u}}{2} d u}}$$

Apply the constant multiple rule $\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$ with $c=\frac{1}{2}$ and $f{\left(u \right)} = e^{u}$:

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

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

$$\frac{\color{red}{\int{e^{u} d u}}}{2} = \frac{\color{red}{e^{u}}}{2}$$

Recall that $u=x^{2}$:

$$\frac{e^{\color{red}{u}}}{2} = \frac{e^{\color{red}{x^{2}}}}{2}$$

Therefore,

$$\int{x e^{x^{2}} d x} = \frac{e^{x^{2}}}{2}$$

$$\int{x e^{x^{2}} d x} = \frac{e^{x^{2}}}{2}+C$$
Answer: $\int{x e^{x^{2}} d x}=\frac{e^{x^{2}}}{2}+C$