Integrate sqrt( (1+x)/(1-x))?

Question

please help

Answer 1

Multiply nominator and denominator by √(1+x)

Integrals with roots:

Consider

Let

Comments

Find ʃ [ (x dx)/√(1 - x²) ].

Let (1 - x²) = u²  ⟹  u = √(1 - x²)

Apply derivative to each side.

- 2x dx = 2u du   ⟹  x dx = - u du.

ʃ [ (x dx)/√(1 - x²) ]  = ʃ [ (- u du)/u ] = - ʃ 1 du = - u + C= - √(1 - x²) + C.

The final solution ʃ √[ (1 + x)/(1 - x) ] dx = sin^{- 1} (x) - √(1 - x²) + C.