Integral of $$$\frac{\tan{\left(y \right)}}{\ln\left(\cos{\left(y \right)}\right)}$$$

Find $$$\int \frac{\tan{\left(y \right)}}{\ln\left(\cos{\left(y \right)}\right)}\, dy$$$.

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

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

Thus,

$${\color{red}{\int{\frac{\tan{\left(y \right)}}{\ln{\left(\cos{\left(y \right)} \right)}} d y}}} = {\color{red}{\int{\left(- \frac{1}{u \ln{\left(u \right)}}\right)d u}}}$$

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

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

Let $$$v=\ln{\left(u \right)}$$$.

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

The integral becomes

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

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

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

Recall that $$$v=\ln{\left(u \right)}$$$:

$$- \ln{\left(\left|{{\color{red}{v}}}\right| \right)} = - \ln{\left(\left|{{\color{red}{\ln{\left(u \right)}}}}\right| \right)}$$

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

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

Therefore,

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

Add the constant of integration:

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

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