integrate by parts arcsinx/(sqrt(1+x)) dx

Question

 Needing help with an integration by parts problem. A hint was given with the problem as follows, "sqrt(1-x^2)=sqrt((1-x)(1+x)). 

Answer 1

∫(arcsinx)(1 + x)^-1/2dx

 

 

u = arcsinx                                       dv = (1 + x)^-1/2dx

du = 1 / √(1 - x²)dx                            v = 2(1 + x)^1/2

 

 

Then the integral is

 

2(1 + x)^1/2arcsinx - 2∫√((1 + x) / (1 - x²))dx

 

2(1 + x)^1/2 arcsinx - 2∫(1 - x)^-1/2 dx

 

 

You can finish up here by taking care of the sub-integral using substitution.