Integral of $$$\frac{1}{a^{2} + x^{2}}$$$ with respect to $$$x$$$

Find $$$\int \frac{1}{a^{2} + x^{2}}\, dx$$$.

Let $$$u=\frac{x}{\left|{a}\right|}$$$.

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

So,

$${\color{red}{\int{\frac{1}{a^{2} + x^{2}} d x}}} = {\color{red}{\int{\frac{\left|{a}\right|}{a^{2} \left(u^{2} + 1\right)} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{\left|{a}\right|}{a^{2}}$$$ and $$$f{\left(u \right)} = \frac{1}{u^{2} + 1}$$$:

$${\color{red}{\int{\frac{\left|{a}\right|}{a^{2} \left(u^{2} + 1\right)} d u}}} = {\color{red}{\frac{\left|{a}\right| \int{\frac{1}{u^{2} + 1} d u}}{a^{2}}}}$$

The integral of $$$\frac{1}{u^{2} + 1}$$$ is $$$\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$$$:

$$\frac{\left|{a}\right| {\color{red}{\int{\frac{1}{u^{2} + 1} d u}}}}{a^{2}} = \frac{\left|{a}\right| {\color{red}{\operatorname{atan}{\left(u \right)}}}}{a^{2}}$$

Recall that $$$u=\frac{x}{\left|{a}\right|}$$$:

$$\frac{\left|{a}\right| \operatorname{atan}{\left({\color{red}{u}} \right)}}{a^{2}} = \frac{\left|{a}\right| \operatorname{atan}{\left({\color{red}{\frac{x}{\left|{a}\right|}}} \right)}}{a^{2}}$$

Therefore,

$$\int{\frac{1}{a^{2} + x^{2}} d x} = \frac{\left|{a}\right| \operatorname{atan}{\left(\frac{x}{\left|{a}\right|} \right)}}{a^{2}}$$

Add the constant of integration:

$$\int{\frac{1}{a^{2} + x^{2}} d x} = \frac{\left|{a}\right| \operatorname{atan}{\left(\frac{x}{\left|{a}\right|} \right)}}{a^{2}}+C$$

$$$\int \frac{1}{a^{2} + x^{2}}\, dx = \frac{\left|{a}\right| \operatorname{atan}{\left(\frac{x}{\left|{a}\right|} \right)}}{a^{2}} + C$$$A