Integral of $$$15 \operatorname{atan}{\left(12 x \right)}$$$

Find $$$\int 15 \operatorname{atan}{\left(12 x \right)}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=15$$$ and $$$f{\left(x \right)} = \operatorname{atan}{\left(12 x \right)}$$$:

$${\color{red}{\int{15 \operatorname{atan}{\left(12 x \right)} d x}}} = {\color{red}{\left(15 \int{\operatorname{atan}{\left(12 x \right)} d x}\right)}}$$

Let $$$u=12 x$$$.

Then $$$du=\left(12 x\right)^{\prime }dx = 12 dx$$$ (steps can be seen »), and we have that $$$dx = \frac{du}{12}$$$.

The integral becomes

$$15 {\color{red}{\int{\operatorname{atan}{\left(12 x \right)} d x}}} = 15 {\color{red}{\int{\frac{\operatorname{atan}{\left(u \right)}}{12} d u}}}$$

Apply the constant multiple rule $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$ with $$$c=\frac{1}{12}$$$ and $$$f{\left(u \right)} = \operatorname{atan}{\left(u \right)}$$$:

$$15 {\color{red}{\int{\frac{\operatorname{atan}{\left(u \right)}}{12} d u}}} = 15 {\color{red}{\left(\frac{\int{\operatorname{atan}{\left(u \right)} d u}}{12}\right)}}$$

For the integral $$$\int{\operatorname{atan}{\left(u \right)} d u}$$$, use integration by parts $$$\int \operatorname{\omega} \operatorname{dv} = \operatorname{\omega}\operatorname{v} - \int \operatorname{v} \operatorname{d\omega}$$$.

Let $$$\operatorname{\omega}=\operatorname{atan}{\left(u \right)}$$$ and $$$\operatorname{dv}=du$$$.

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

Thus,

$$\frac{5 {\color{red}{\int{\operatorname{atan}{\left(u \right)} d u}}}}{4}=\frac{5 {\color{red}{\left(\operatorname{atan}{\left(u \right)} \cdot u-\int{u \cdot \frac{1}{u^{2} + 1} d u}\right)}}}{4}=\frac{5 {\color{red}{\left(u \operatorname{atan}{\left(u \right)} - \int{\frac{u}{u^{2} + 1} d u}\right)}}}{4}$$

Let $$$v=u^{2} + 1$$$.

Then $$$dv=\left(u^{2} + 1\right)^{\prime }du = 2 u du$$$ (steps can be seen »), and we have that $$$u du = \frac{dv}{2}$$$.

Thus,

$$\frac{5 u \operatorname{atan}{\left(u \right)}}{4} - \frac{5 {\color{red}{\int{\frac{u}{u^{2} + 1} d u}}}}{4} = \frac{5 u \operatorname{atan}{\left(u \right)}}{4} - \frac{5 {\color{red}{\int{\frac{1}{2 v} d v}}}}{4}$$

Apply the constant multiple rule $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$ with $$$c=\frac{1}{2}$$$ and $$$f{\left(v \right)} = \frac{1}{v}$$$:

$$\frac{5 u \operatorname{atan}{\left(u \right)}}{4} - \frac{5 {\color{red}{\int{\frac{1}{2 v} d v}}}}{4} = \frac{5 u \operatorname{atan}{\left(u \right)}}{4} - \frac{5 {\color{red}{\left(\frac{\int{\frac{1}{v} d v}}{2}\right)}}}{4}$$

The integral of $$$\frac{1}{v}$$$ is $$$\int{\frac{1}{v} d v} = \ln{\left(\left|{v}\right| \right)}$$$:

$$\frac{5 u \operatorname{atan}{\left(u \right)}}{4} - \frac{5 {\color{red}{\int{\frac{1}{v} d v}}}}{8} = \frac{5 u \operatorname{atan}{\left(u \right)}}{4} - \frac{5 {\color{red}{\ln{\left(\left|{v}\right| \right)}}}}{8}$$

Recall that $$$v=u^{2} + 1$$$:

$$\frac{5 u \operatorname{atan}{\left(u \right)}}{4} - \frac{5 \ln{\left(\left|{{\color{red}{v}}}\right| \right)}}{8} = \frac{5 u \operatorname{atan}{\left(u \right)}}{4} - \frac{5 \ln{\left(\left|{{\color{red}{\left(u^{2} + 1\right)}}}\right| \right)}}{8}$$

Recall that $$$u=12 x$$$:

$$- \frac{5 \ln{\left(1 + {\color{red}{u}}^{2} \right)}}{8} + \frac{5 {\color{red}{u}} \operatorname{atan}{\left({\color{red}{u}} \right)}}{4} = - \frac{5 \ln{\left(1 + {\color{red}{\left(12 x\right)}}^{2} \right)}}{8} + \frac{5 {\color{red}{\left(12 x\right)}} \operatorname{atan}{\left({\color{red}{\left(12 x\right)}} \right)}}{4}$$

Therefore,

$$\int{15 \operatorname{atan}{\left(12 x \right)} d x} = 15 x \operatorname{atan}{\left(12 x \right)} - \frac{5 \ln{\left(144 x^{2} + 1 \right)}}{8}$$

Add the constant of integration:

$$\int{15 \operatorname{atan}{\left(12 x \right)} d x} = 15 x \operatorname{atan}{\left(12 x \right)} - \frac{5 \ln{\left(144 x^{2} + 1 \right)}}{8}+C$$

$$$\int 15 \operatorname{atan}{\left(12 x \right)}\, dx = \left(15 x \operatorname{atan}{\left(12 x \right)} - \frac{5 \ln\left(144 x^{2} + 1\right)}{8}\right) + C$$$A