How to determine the antiderivative of the function xarcsin

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

Solution

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.