Inverse sine trig sub integral arcsin(sqrt(1-x^2)), trig sub with integration by parts integral.

In this inverse sine trig sub integral, we compute the integral arcsin(sqrt(1-x^2) using trig substitution with integration by parts.

First, we choose the substitution that will turn sqrt(1-x^2) into sin(theta): let x=cos(theta), apply a pythagorean identity and simplify the square root. The inverse sine undoes the sine, and that part of the trig sub integrand is simplified. We also transform the differential to complete the trig sub transformation, and this gives us the classic integration by parts integral theta*sin(theta).

We let u=theta and dv=sin(theta)d(theta) and apply the integration by parts formula. Next, we compute the final integral for integration by parts to obtain the antiderivative in terms of theta.

Now we have to transform the antiderivative from theta to x, and we use a reference triangle for this. We draw a triangle with the angle theta=arccos(x) and solve for the missing side, and this allows us to compute sin(arccos(x)) and express the final answer in terms of x.