Integral of the equation

Question

∫1/(√x√(1-x)) dx

Answer 1

This is a little tricky, but not too bad.  Let u = √x, then x = u² and dx = 2u du.

 

The original integral can then be written as:

 

∫ 1/(u√(1 - u²)) 2u du = ∫ 2 du/√(1 - u²) = 2 ∫ du/√(1 - u²) = 2 sin^-1 u + C = 2 sin^-1 √x + C.

 

The answer can be checked by differentiating and simplifying to the original integrand.