How do you evaluate the definite integral int 2(pi)x(cos^(-1)(x))dx from 0 to 1?
1 Answer
Explanation:
Before beginning, we should know or determine that
Working first with the unbounded integral, we should apply integration by parts. Let:
{(u=cos^-1x,=>,du=(-1)/sqrt(1-x^2)dx),(dv=xdx,=>,v=x^2/2):}
Then:
On the remaining integral, let
Simplifying:
=1/2theta-1/2sinthetacostheta=1/2sin^-1x-1/2xsqrt(1-x^2)
Plugging this into our previous expression:
Now applying the bounds, the original integral equals:
=picos^-1(1)+pi/2sin^-1(1)-(pi/2sin^-1(0))
=pi/2(pi/2)=pi^2/4