Integral of $$$f x^{a}$$$ with respect to $$$x$$$

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

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

$${\color{red}{\int{f x^{a} d x}}} = {\color{red}{f \int{x^{a} d x}}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=a$$$:

$$f {\color{red}{\int{x^{a} d x}}}=f {\color{red}{\frac{x^{a + 1}}{a + 1}}}=f {\color{red}{\frac{x^{a + 1}}{a + 1}}}$$

Therefore,

$$\int{f x^{a} d x} = \frac{f x^{a + 1}}{a + 1}$$

Add the constant of integration:

$$\int{f x^{a} d x} = \frac{f x^{a + 1}}{a + 1}+C$$

$$$\int f x^{a}\, dx = \frac{f x^{a + 1}}{a + 1} + C$$$A