Let ∫ dx/(√(x+1) - √(x-1)) = α(x+1)^(3/2) + β(x-1)^(3/2) + c. What is the value of α?

Rationalize: 1/(√(x+1) - √(x-1)) × (√(x+1) + √(x-1))/(√(x+1) + √(x-1)) = (√(x+1) + √(x-1))/((x+1) - (x-1)) = (√(x+1) + √(x-1))/2. So integral = (1/2)∫[√(x+1) + √(x-1)] dx = (1/2)[(2/3)(x+1)^(3/2) + (2/3)(x-1)^(3/2)] + c = (1/3)(x+1)^(3/2) + (1/3)(x-1)^(3/2) + c. So α = 1/3.