Question

evaluate the integral using any appropriate algebraic method or trigonometric identity.
int\frac{3mathrm{d}x}{xsqrt{4 - 9ln^{2}x}}
int\frac{3mathrm{d}x}{xsqrt{4 - 9ln^{2}x}}=square (type an exact answer.)

Explanation:

Step1: Use substitution

Let $u = 3\ln x$, then $du=\frac{3}{x}dx$.

Step2: Rewrite integral

The integral becomes $\int\frac{du}{\sqrt{4 - u^{2}}}$.

Step3: Recall integral formula

$\int\frac{du}{\sqrt{a^{2}-u^{2}}}=\arcsin(\frac{u}{a})+C$, here $a = 2$.

Step4: Substitute back

We get $\arcsin(\frac{3\ln x}{2})+C$.

Answer:

$\arcsin(\frac{3\ln x}{2})+C$