How to determine the antiderivative of the function x*arcsin x/square root(1-x^2)?

To determine the antiderivative of a given function, we'll
have to calculate the indefinite integral of that
function.

We'll apply integration by parts. First, we'll
recall the formula:

Int udv = u*v - Int
vdu

Let u = arcsin x => du =
dx/sqrt(1-x^2)

Let dv = xdx/sqrt(1-x^2) => v =
-sqrt(1 - x^2)

Int x*arcsin x dx/sqrt(1-x^2) = -(arcsin
x)*sqrt(1 - x^2) + Intsqrt(1 - x^2)dx/sqrt(1-x^2)

Int
x*arcsin x dx/sqrt(1-x^2) = -(arcsin x)*sqrt(1 - x^2) + Int
dx

Int x*arcsin x dx/sqrt(1-x^2) = -(arcsin x)*sqrt(1 -
x^2) + x + C

The antiderivative of the given
function f(x) is: F(x) = -(arcsin x)*sqrt(1 - x^2) + x +
C.