find the indefinite integral. (use c for the constant of integration.)\n int\frac{1}{xsqrt{9x^{2}-1}}dx

Explanation:

Step1: Use substitution

Let $u = 3x$, then $du=3dx$ and $x=\frac{u}{3}$. The integral $\int\frac{1}{x\sqrt{9x^{2}-1}}dx=\int\frac{1}{\frac{u}{3}\sqrt{u^{2}-1}}\cdot\frac{du}{3}=\int\frac{1}{u\sqrt{u^{2}-1}}du$.

Step2: Recall the standard - integral formula

The antiderivative of $\frac{1}{u\sqrt{u^{2}-1}}$ is $\text{arcsec}(|u|)+C$. Since $u = 3x$, the result of the original integral is $\text{arcsec}(|3x|)+C$.

Answer:

$\text{arcsec}(|3x|)+C$