Integral of $$$a^{\frac{x}{b}}$$$ with respect to $$$x$$$

Find $$$\int a^{\frac{x}{b}}\, dx$$$.

Let $$$u=\frac{x}{b}$$$.

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

Therefore,

$${\color{red}{\int{a^{\frac{x}{b}} d x}}} = {\color{red}{\int{a^{u} b d u}}}$$

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

$${\color{red}{\int{a^{u} b d u}}} = {\color{red}{b \int{a^{u} d u}}}$$

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

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

Recall that $$$u=\frac{x}{b}$$$:

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

Therefore,

$$\int{a^{\frac{x}{b}} d x} = \frac{a^{\frac{x}{b}} b}{\ln{\left(a \right)}}$$

Add the constant of integration:

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

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