use integration by parts to evaluate the indefinite integral ∫(arcsinx)/sqrt(1+x)

Answer 1

Let u = arcsin(x)     and vdv = dx/sqrt(1+x)    so v = 2 sqrt(1+x)

By parts, the integral is:  2 sqrt(1+x) arcsin(x) – integral [2 sqrt((1+x)/(1 – x^2)) ]

By factoring 1 – x^2   this is seen to be

2 sqrt(1+x) arcsin(x) – integral [ 2 /sqrt(1-x)   ]    =

2 sqrt((1+x) arcsin(x) + 4 sqrt(1-x)

Then add the constant of integral to get  2 sqrt((1+x) arcsin(x) + 4 sqrt(1-x) + C