Integral of $$$a f - b f$$$ with respect to $$$a$$$

Find $$$\int \left(a f - b f\right)\, da$$$.

Integrate term by term:

$${\color{red}{\int{\left(a f - b f\right)d a}}} = {\color{red}{\left(\int{a f d a} - \int{b f d a}\right)}}$$

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

$$- \int{b f d a} + {\color{red}{\int{a f d a}}} = - \int{b f d a} + {\color{red}{f \int{a d a}}}$$

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

$$f {\color{red}{\int{a d a}}} - \int{b f d a}=f {\color{red}{\frac{a^{1 + 1}}{1 + 1}}} - \int{b f d a}=f {\color{red}{\left(\frac{a^{2}}{2}\right)}} - \int{b f d a}$$

Apply the constant rule $$$\int c\, da = a c$$$ with $$$c=b f$$$:

$$\frac{a^{2} f}{2} - {\color{red}{\int{b f d a}}} = \frac{a^{2} f}{2} - {\color{red}{a b f}}$$

Therefore,

$$\int{\left(a f - b f\right)d a} = \frac{a^{2} f}{2} - a b f$$

Simplify:

$$\int{\left(a f - b f\right)d a} = \frac{a f \left(a - 2 b\right)}{2}$$

Add the constant of integration:

$$\int{\left(a f - b f\right)d a} = \frac{a f \left(a - 2 b\right)}{2}+C$$

$$$\int \left(a f - b f\right)\, da = \frac{a f \left(a - 2 b\right)}{2} + C$$$A