Evaluate integral01 (1)/(sqrt(9x2 - 1)) dx

Here are the solutions to the integration problems.

:
Evaluate $\int_0^1 \frac{1}{\sqrt{9x^2 - 1}} \, dx$

Assumption: The integrand $\frac{1}{\sqrt{9x^2 - 1}}$ is only defined for $9x^2 - 1 > 0$, which means $x^2 > \frac{1}{9}$, or $x > \frac{1}{3}$ or $x < -\frac{1}{3}$. Since the lower limit of integration is 0, the integral as written is improper and undefined in real numbers over the interval $[0, 1]$. We will evaluate the integral from $x = \frac{1}{3}$ to $x