my question has to do with mixed integrals

Question

how do you take the integral of

∫1/(x√x2−4 dx)

Answer 1

desmos.com/calculator/lhbbhelqmd

You might want to watch some videos on using trig substitution. Try a search for "professor Leonard" and trig substitution.

Answer 2

Let x = 2secθ. Then dx = 2secθtanθdθ

Substitute into the integral to get: ∫[1 / (2secθ√(4sec²θ - 4))] 2secθtanθdθ

= (1/2)∫ (secθtanθ / [(secθ√(sec²θ - 1)] dθ = (1/2)∫(secθtanθ / (secθtanθ) dθ = (1/2)∫dθ =

(1/2)θ + C Since secθ = x/2, θ = Sec^-1(x/2).

So, the answer is (1/2)Sec^-1(x/2) + C