Question

Evaluate $\int \frac{\sqrt{x^2-a^2}}{x}dx$

Answer 1

put $x=asec\theta$

$dx=atan\theta sec\theta d\theta$

$I=\int \frac{atan\theta }{asec\theta }atan\theta sec\theta d\theta$

$=\int a(se{c}^2\theta -1)d\theta$

$=a(tan\theta -\theta )+c$

$I=a\left[\sqrt{(\frac{x^2}{a^2}-1}\right)-\left(se{c}^{-1}\frac{x}{a}\right)]+c$