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

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

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

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

Rewrite the tangent as $$$\tan\left(x\right)=\frac{\sin\left(x\right)}{\cos\left(x\right)}$$$:

$$\frac{{\color{red}{\int{\tan{\left(x \right)} d x}}}}{2} = \frac{{\color{red}{\int{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} d x}}}}{2}$$

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

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

Thus,

$$\frac{{\color{red}{\int{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} d x}}}}{2} = \frac{{\color{red}{\int{\left(- \frac{1}{u}\right)d u}}}}{2}$$

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

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

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

$$- \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{2} = - \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{2}$$

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

$$- \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{2} = - \frac{\ln{\left(\left|{{\color{red}{\cos{\left(x \right)}}}}\right| \right)}}{2}$$

Therefore,

$$\int{\frac{\tan{\left(x \right)}}{2} d x} = - \frac{\ln{\left(\left|{\cos{\left(x \right)}}\right| \right)}}{2}$$

Add the constant of integration:

$$\int{\frac{\tan{\left(x \right)}}{2} d x} = - \frac{\ln{\left(\left|{\cos{\left(x \right)}}\right| \right)}}{2}+C$$

$$$\int \frac{\tan{\left(x \right)}}{2}\, dx = - \frac{\ln\left(\left|{\cos{\left(x \right)}}\right|\right)}{2} + C$$$A