Question

Integration of 1+tanx/x+log(sec(x))?

Answer 1

The answer is `log|x+log(secx)|`

Explanation:

We perform this integration like this

`int (1+tanx)/(x+log(secx))dx`

Using substitution method :
1) Let x + log(secx) = t
2) On differentiating it with respect to x you get

`1+(1/secx*secx.tanx)=dt/dx`

This has become so because derivative of x w.r.t to x is 1 , derivative of log x is 1/x and chain rule is also used.

3) Simplify the above step by cancelling secx , and take dx to the LHS , you get

`(1+tanx)dx = dt`

Now (1+tanx)dx = dt and x + log(secx) = t
Substitute these into the original integral , you get ,

->`intdt/t`

On integration , `intdt/t=log|t|`

Substitute t as x+log(secx)
and your answer is ,

`log|x+log(secx)|+c`