integration for x>5

Question

I want to integrate (x^2-25)^0.5 / x What method is best to use for this? Also if it doesn't state values for the integral, just that x>5, is there something in particular that needs to be done to accommodate this? thanks

Answer 1

Using Trigonometric substitution, $$x=5\sec\theta$$

$$\int\frac{\sqrt{x^2-25}}xdx=\int \frac{5|\tan\theta|}{5\sec\theta}5\sec\theta\tan\theta d\theta$$

$$=5\cdot\text{sgn}(\tan\theta)\int\tan^2\theta d\theta=5\cdot\text{sgn}(\tan\theta)\int(\sec^2\theta-1) d\theta$$