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

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

Let $$$u=x^{2}$$$.

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

The integral becomes

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

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

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

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

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

Therefore,

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

Add the constant of integration:

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

$$$\int 2 x e^{x^{2}}\, dx = e^{x^{2}} + C$$$A