Integrate the function f(x) and compare the given solution to find the value of the constants 'a' and 'b'

Question

If int {1/x√(1-(x)^3) } dx = a log mod {√(1-x^3-1) / √(1- x^3 +1) } + b ,

Then find the value of 'a' and 'b' .

Answer 1

If you review your inverse hyperbolic functions, you find that

tanh^-1x = 1/2 log|(1+x)/(1-x)|

That is not quite what you are given. The fraction is upside down, so

-tanh^-1x = 1/2 log|(x-1)/(x+1)|

Now replace x with √(1-x^3) and after some algebra you find that the given integral is

-2/3 tanh^-1√(1-x³) + C = 1/3 log|(√(1-x³)-1)/(√(1-x³)+1)| + C

So I guess a = 1/3 and b cannot be determined.