Question

Integration of ; ((1/X(4 + Inx))dx?

Answer 1

`int1/(x(4+lnx))dx=ln|4+lnx|+C`

Explanation:

So, we want to determine

`int1/(x(4+lnx))dx`

This can be solved using a substitution:

`u=lnx`

`du=dx/x`

Rewriting the integral a little, we see that `du` indeed shows up:

`int1/(4+lnx)*dx/x`

So, apply the substitution:

`int(du)/(4+u)`

We can apply a second brief substitution here:

`v=4+u`

`dv=du`

`int(dv)/v=ln|v|+C`

`=ln|4+u|+C`

Rewrite in terms of `x:`

`int1/(x(4+lnx))dx=ln|4+lnx|+C`