U sub instead of trig sub

Question

indefinite integral of sqrt((4-x)/x)

solve when u= sqrt(x/(4-x)) then your dx= 1/[(x^1/2/(2(4-x)^3/2)+(1/(2(4-x)^1/2(x)^1/2))]

then you get integral 1/(u^2+1) with an 8 outside the integral sign

show me steps to get that if possible please

Answer 1

∫ √(4-x)/√(x) dx

x = 4 sin²θ

dx = 8sin θ * cos θ dθ

∫ √(4 - 4sin²θ)/√(4sin²θ) * 8sinθ*cosθ dθ =

∫(cos θ)/(sin θ) *8sin θ * cos θ dθ

∫8cos²θ dθ cos²θ = (1+cos 2θ)/2

∫4 + 4cos 2θ dθ = 4θ + 2sin 2θ

We now set up a right triangle based on the original parameter x = 4sin²θ

sin²θ = x/4

sin θ = √x/2

cos θ = √(4 - x)/2√

θ = sin^-1(√x/2)

4sin^-1(√x/2) + √x√(4-x) + C