Question

Ex / 1 - e2x d x solving this integral.

Answer 1

`I=(1)/(2)(\ln \left|e^x+1\right|)/(\ln \left|e^x-1\right|)+C`

Explanation:

The given in integral problem is

`\int (e^x)/(1-e^(2x))dx`

`\int (e^x)/(1-(e^(x))^2)dx`

Substitute
`e^x=t`.

`(d)/(dx)e^x=(d)/(dx)t`

`e^x=(dt)/(dx)`

`e^xdx=dt`

After substitution we get

`\int (t)/(1-t^2)dt`

We know that

`\int (1)/(a^2-x^2)dc=(1)/(2a)(\ln \left|x+a\right|)/(\ln \left|x-a\right|)+C`

Using this formula we get

`\int (1)/(1^2-t^2)du=(1)/(2(1))(\ln \left|t+1\right|)/(\ln \left|t-1\right|)+C`

Substitute
`t=e^x`.

`I=(1)/(2)(\ln \left|e^x+1\right|)/(\ln \left|e^x-1\right|)+C`

Therefore, the solution of given integral is
`I=(1)/(2)(\ln \left|e^x+1\right|)/(\ln \left|e^x-1\right|)+C`.