Question

Find the integral of the function $\frac{tan\left(3ln\right(x\left)\right)}{x}$

• A.
• B.
• C.
• D.

Answer 1

The correct option is D

We saw that $\int tanxdx=ln|secx|+c$
We have lnx and it’s derivative $\frac{1}{x}$ in the function we want to integrate. So the substitution we make would be,
3 lnx = u.
(If we substitute lnx = u, we would need one more substitution to account for 3 inside tan function(or more steps).
$\Rightarrow 3lnx=u\Rightarrow du=\frac{3}{x}dx\Rightarrow \frac{1}{x}dx=\frac{du}{3}$
So the integral $\int \frac{tan\left(3ln\right(x\left)\right)}{x}dx$ becomes $\int tan\left(u\right)\frac{du}{3}$
This will be equal to $\frac{ln|sec\left(u\right)|}{3}$
Replacing u with 3 lnx, we get $\frac{ln|sec\left(3lnx\right)|}{3}$
$\Rightarrow \int \frac{tan\left(3ln\right(x\left)\right)}{x}dx=\frac{ln|sec\left(3lnx\right)|}{3}+c$