Okay, so we have
[tex] \int\limits { \dfrac{7ln(x)}{x \sqrt{5+ln(x)^2}}} \, dx [/tex].
Substitute [tex]u=ln(x)^2+5[/tex] and differentiate.
[tex]dx= \dfrac{x}{2ln(x)}du[/tex]
[tex] \dfrac{7}{2} \int\limits { \dfrac{1}{ \sqrt{u}}} \, du [/tex]
[tex]\int\limits { \dfrac{1}{ \sqrt{u}}} \, du = 2 \sqrt{u} [/tex]
[tex] \dfrac{7}{2} \int\limits { \dfrac{1}{ \sqrt{u}} } \, du = 7 \sqrt{ln(x)^2+5}+C [/tex]
