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

Apr 29, 2018

$\int \frac{1}{x \left(4 + \ln x\right)} \mathrm{dx} = \ln | 4 + \ln x | + C$

#### Explanation:

So, we want to determine

$\int \frac{1}{x \left(4 + \ln x\right)} \mathrm{dx}$

This can be solved using a substitution:

$u = \ln x$

$\mathrm{du} = \frac{\mathrm{dx}}{x}$

Rewriting the integral a little, we see that $\mathrm{du}$ indeed shows up:

$\int \frac{1}{4 + \ln x} \cdot \frac{\mathrm{dx}}{x}$

So, apply the substitution:

$\int \frac{\mathrm{du}}{4 + u}$

We can apply a second brief substitution here:

$v = 4 + u$

$\mathrm{dv} = \mathrm{du}$

$\int \frac{\mathrm{dv}}{v} = \ln | v | + C$

$= \ln | 4 + u | + C$

Rewrite in terms of $x :$

$\int \frac{1}{x \left(4 + \ln x\right)} \mathrm{dx} = \ln | 4 + \ln x | + C$