Evaluate the integral ∫dxx3x2−1\int \frac{dx}{x^3 \sqrt{x^2-1}}∫x3x2−1dx using trigonometric substitution.
x2−1x2+C\frac{\sqrt{x^2-1}}{x^2} + Cx2x2−1+C
12x2−1+C\frac{1}{2}\sqrt{x^2-1} + C21x2−1+C
12arcsec(x)+x2−12x2+C\frac{1}{2}\text{arcsec}(x) + \frac{\sqrt{x^2-1}}{2x^2} + C21arcsec(x)+2x2x2−1+C
12arcsec(x)−x2−12x2+C\frac{1}{2}\text{arcsec}(x) - \frac{\sqrt{x^2-1}}{2x^2} + C21arcsec(x)−2x2x2−1+C