Calculus of a Single Variable, Chapter 8, 8.4, Section 8.4, Problem 36

Recall that indefinite integral follows int f(x) dx = F(x) +C where:
f(x) as the integrand function
F(x) as the antiderivative of f(x)
C as the constant of integration.
The given integral problem: int x arcsin(x)dx resembles a formula from integration table. We follow the integral formula for inverse sine function as:
int x arcsin(ax) dx = (x^2arcsin(ax))/2-arcsin(ax)/(4a^2)+(xsqrt(1-a^2x^2))/(4a)+C
Applying the integral formula inverse sine function with a=1 , we get:
int x arcsin(x) dx = (x^2arcsin(1*x))/2-arcsin(1*x)/(4*1^2)+(xsqrt(1-1^2x^2))/(4*1)+C
= (x^2arcsin(x))/2-arcsin(x)/4+(xsqrt(1-x^2))/4+C
or ((2x^2-1)arcsin(x)+xsqrt(1-x^2))/4+C