Integral of $$$5^{x^{2}} x$$$

Find $$$\int 5^{x^{2}} x\, dx$$$.

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

$${\color{red}{\int{5^{x^{2}} x d x}}} = {\color{red}{\int{\frac{5^{u}}{2} d u}}}$$

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 $$$f{\left(u \right)} = 5^{u}$$$:

$${\color{red}{\int{\frac{5^{u}}{2} d u}}} = {\color{red}{\left(\frac{\int{5^{u} d u}}{2}\right)}}$$

Apply the exponential rule $$$\int{a^{u} d u} = \frac{a^{u}}{\ln{\left(a \right)}}$$$ with $$$a=5$$$:

$$\frac{{\color{red}{\int{5^{u} d u}}}}{2} = \frac{{\color{red}{\frac{5^{u}}{\ln{\left(5 \right)}}}}}{2}$$

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

$$\frac{5^{{\color{red}{u}}}}{2 \ln{\left(5 \right)}} = \frac{5^{{\color{red}{x^{2}}}}}{2 \ln{\left(5 \right)}}$$

Therefore,

$$\int{5^{x^{2}} x d x} = \frac{5^{x^{2}}}{2 \ln{\left(5 \right)}}$$

Add the constant of integration:

$$\int{5^{x^{2}} x d x} = \frac{5^{x^{2}}}{2 \ln{\left(5 \right)}}+C$$

$$$\int 5^{x^{2}} x\, dx = \frac{5^{x^{2}}}{2 \ln\left(5\right)} + C$$$A