Question

Integral Ln|x|/(1+x)^2 dx Answer Guys ???

Answer 1

Use integration by parts.

Explanation:

Let

`I=int(ln|x|)/(1+x)^2dx`

Apply integration by parts:

`u(x)=ln|x|`, `u'(x)=1/x`.
`v'(x)=1/(1+x)^2`, `v(x)=-1/(1+x)`.

Hence

`I=-(ln|x|)/(1+x)+int1/(x(1+x))dx`

Apply partial fraction decomposition:

`I=-(ln|x|)/(1+x)+int(1/x-1/(1+x))dx`

Integrate term by term:

`I=-(ln|x|)/(1+x)+ln|x|-ln|1+x|+C`

Simplify:

`I=x/(1+x)ln|x|-ln|1+x|+C`