Question

```
some integrals... I will be adding more and more problems, as times goes on.
Of course at some point this question will end.
```

Answer 1

NUMBER 1.
\(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\sec x~dx}\)

\(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{1 \cdot\cos x}{\cos x\cdot\cos x}~dx}\)

\(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{\cos x}{1-\sin^2 x}~dx}\)

\(\large\color{slate}{\displaystyle u=\sin x}\)
\(\large\color{slate}{\displaystyle du=\cos x~dx}\)


\(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\frac{1}{1-u^2}~du}\)

\(\large\color{slate}{\displaystyle\frac{1}{2}\int\limits_{~}^{~}\frac{1}{1-u}+\frac{1}{1+u}~du}\)

\(\large\color{slate}{\displaystyle\frac{1}{2}\int\limits_{~}^{~}\frac{1}{u+1}-\frac{1}{u-1}~du}\)

\(\large\color{slate}{\displaystyle\frac{1}{2}{\rm Ln}\left( \frac{\sin x+1}{\sin x-1} \right)+C}\)

\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, royalblue ,border:2px solid royalblue ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)

Answer 2

NUMBER 2.
\(\large\color{slate}{\displaystyle\int\limits_{~}^{~}\cos x~\ln(\tan x)~dx}\)

\(\large\color{slate}{\displaystyle u=\ln(\tan x)}\)\(\Huge\color{white}{ \left| \right| }\)
\(\large\color{slate}{\displaystyle du=\frac{\sec^2x }{\tan x}=\frac{1 }{\sin x~\cos x}}\)\(\Huge\color{white}{ \left| \right| }\)


\(\large\color{slate}{\displaystyle dv=\cos x}\) \(\large\color{slate}{\displaystyle v=\sin x}\)
\(\Huge\color{white}{ \left| \right| }\)
\(\large\color{slate}{\displaystyle\sin x~\ln(\tan x)-\int\limits_{~}^{~}\frac{\sin x}{\sin x~\cos x}~dx}\)

\(\Huge\color{white}{ \left| \right| }\)
\(\large\color{slate}{\displaystyle\sin x~\ln(\tan x)+\frac{1}{2}\ln\left(\frac{\sin x-1}{\sin x+1} \right)+C}\)

\(\scriptsize\color{ slate }{\scriptsize{\bbox[5pt, royalblue ,border:2px solid royalblue ]{~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ }}}\)

Answer 3

(again, I will depart from thsi question, but I will be adding more problems)
again, don't reply....