Evaluate the integral integral 8 sin1 x dxSolutionThe first

Evaluate the integral integral -8 sin^-1 (x) dx

Solution

The first step to integrate the function f(x) = arcsin(x) is to apply the standard trick of integration by parts. The integration by parts formula is

u * dv = u * v - v * du

In the integral arcsin(x) dx, we let u = arcsin(x), dx = dv, du = 1/sqrt(1 - x^2) dx, and v = x. This gives us the integral equation

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

The right-hand side of this equation contains a new integral that looks slightly complicated, but we can still work it out if we apply the trick of substitution. Let\'s set w = x^2 and dw = 2x dx. Now we get the integral equivalence

x/sqrt(1 - x^2) dx
= (1/2) * 1/sqrt(1 - w) dw
= -sqrt(1 - w) + c
= -sqrt(1 - x^2) + c

If we put all the separate pieces together, we get the final expression for the antiderivative of the function y = arcsin(x).

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

= -8*(x*arcsin(x) + sqrt(1 - x^2) + c)