Question

Evaluate the integral. (use c for the constant of integration.) 7 ln(x)/ x sqrt(5 + (ln(x))^2) dx

Answer 1

Okay, so we have 

$\int\limits { \dfrac{7ln(x)}{x \sqrt{5+ln(x)^2}}} \, dx$.

Substitute $u=ln(x)^2+5$ and differentiate.
$dx= \dfrac{x}{2ln(x)}du$
$\dfrac{7}{2} \int\limits { \dfrac{1}{ \sqrt{u}}} \, du$

$\int\limits { \dfrac{1}{ \sqrt{u}}} \, du = 2 \sqrt{u}$

$\dfrac{7}{2} \int\limits { \dfrac{1}{ \sqrt{u}} } \, du = 7 \sqrt{ln(x)^2+5}+C$