Prove that: ∫dx/(x + (a^2 - x^2)) for x ∈ [0,a] = π/4

Question

Prove that: \(\int_0^a \mathrm{dx}/(x + \sqrt {a^2 - x^2})\,\mathrm{d}x\)

Answer 1

Consider x = a sin t

We get dx = a cos t dt

Here if x = 0 we get t = 0 

Consider sin t + cos t = z

We get (cos t - sin t) dt = dz

Here if t = 0 we get z = 1