Integral of $$$9 \tan^{2}{\left(x \right)}$$$

Find $$$\int 9 \tan^{2}{\left(x \right)}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=9$$$ and $$$f{\left(x \right)} = \tan^{2}{\left(x \right)}$$$:

$${\color{red}{\int{9 \tan^{2}{\left(x \right)} d x}}} = {\color{red}{\left(9 \int{\tan^{2}{\left(x \right)} d x}\right)}}$$

Let $$$u=\tan{\left(x \right)}$$$.

Then $$$x=\operatorname{atan}{\left(u \right)}$$$ and $$$dx=\left(\operatorname{atan}{\left(u \right)}\right)^{\prime }du = \frac{du}{u^{2} + 1}$$$ (steps can be seen »).

Therefore,

$$9 {\color{red}{\int{\tan^{2}{\left(x \right)} d x}}} = 9 {\color{red}{\int{\frac{u^{2}}{u^{2} + 1} d u}}}$$

Rewrite and split the fraction:

$$9 {\color{red}{\int{\frac{u^{2}}{u^{2} + 1} d u}}} = 9 {\color{red}{\int{\left(1 - \frac{1}{u^{2} + 1}\right)d u}}}$$

Integrate term by term:

$$9 {\color{red}{\int{\left(1 - \frac{1}{u^{2} + 1}\right)d u}}} = 9 {\color{red}{\left(\int{1 d u} - \int{\frac{1}{u^{2} + 1} d u}\right)}}$$

Apply the constant rule $$$\int c\, du = c u$$$ with $$$c=1$$$:

$$- 9 \int{\frac{1}{u^{2} + 1} d u} + 9 {\color{red}{\int{1 d u}}} = - 9 \int{\frac{1}{u^{2} + 1} d u} + 9 {\color{red}{u}}$$

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

$$9 u - 9 {\color{red}{\int{\frac{1}{u^{2} + 1} d u}}} = 9 u - 9 {\color{red}{\operatorname{atan}{\left(u \right)}}}$$

Recall that $$$u=\tan{\left(x \right)}$$$:

$$- 9 \operatorname{atan}{\left({\color{red}{u}} \right)} + 9 {\color{red}{u}} = - 9 \operatorname{atan}{\left({\color{red}{\tan{\left(x \right)}}} \right)} + 9 {\color{red}{\tan{\left(x \right)}}}$$

Therefore,

$$\int{9 \tan^{2}{\left(x \right)} d x} = 9 \tan{\left(x \right)} - 9 \operatorname{atan}{\left(\tan{\left(x \right)} \right)}$$

Simplify:

$$\int{9 \tan^{2}{\left(x \right)} d x} = - 9 x + 9 \tan{\left(x \right)}$$

Add the constant of integration:

$$\int{9 \tan^{2}{\left(x \right)} d x} = - 9 x + 9 \tan{\left(x \right)}+C$$

$$$\int 9 \tan^{2}{\left(x \right)}\, dx = \left(- 9 x + 9 \tan{\left(x \right)}\right) + C$$$A