Take the integral:
integral u^2 sqrt(a^2 - u^2) du
For the integrand u^2 sqrt(a^2 - u^2), (assuming all variables are positive) substitute u = a sin(u) and du = a cos(u) du. Then sqrt(a^2 - u^2) = sqrt(a^2 - a^2 sin^2(u)) = a cos(u) and u = sin^(-1)(u/a):
= a integral a^3 sin^2(u) cos^2(u) du
Factor out constants:
= a^4 integral sin^2(u) cos^2(u) du
Write cos^2(u) as 1 - sin^2(u):
= a^4 integral sin^2(u) (1 - sin^2(u)) du
Expanding the integrand sin^2(u) (1 - sin^2(u)) gives sin^2(u) - sin^4(u):
= a^4 integral(sin^2(u) - sin^4(u)) du
Integrate the sum term by term and factor out constants:
= -a^4 integral sin^4(u) du + a^4 integral sin^2(u) du
Use the reduction formula, integral sin^m(u) du = -(cos(u) sin^(m - 1)(u))/m + (m - 1)/m integral sin^(-2 + m)(u) du, where m = 4:
= 1/4 a^4 sin^3(u) cos(u) + a^4/4 integral sin^2(u) du
Write sin^2(u) as 1/2 - 1/2 cos(2 u):
= 1/4 a^4 sin^3(u) cos(u) + a^4/4 integral(1/2 - 1/2 cos(2 u)) du
Integrate the sum term by term and factor out constants:
= 1/4 a^4 sin^3(u) cos(u) - a^4/8 integral cos(2 u) du + a^4/8 integral1 du
For the integrand cos(2 u), substitute s = 2 u and ds = 2 du:
= 1/4 a^4 sin^3(u) cos(u) - a^4/16 integral cos(s) ds + a^4/8 integral1 du
The integral of cos(s) is sin(s):
= -1/16 a^4 sin(s) + 1/4 a^4 sin^3(u) cos(u) + a^4/8 integral1 du
The integral of 1 is u:
= -1/16 a^4 sin(s) + (a^4 u)/8 + 1/4 a^4 sin^3(u) cos(u) + constant
Substitute back for s = 2 u:
= (a^4 u)/8 - 1/16 a^4 sin(2 u) + 1/4 a^4 sin^3(u) cos(u) + constant
Apply the double angle formula sin(2 u) = 2 sin(u) cos(u):
= (a^4 u)/8 + 1/4 a^4 sin^3(u) cos(u) - 1/8 a^4 sin(u) cos(u) + constant
Express cos(u) in terms of sin(u) using cos^2(u) = 1 - sin^2(u):
= (a^4 u)/8 - 1/8 a^4 sin(u) sqrt(1 - sin^2(u)) + 1/4 a^4 sin^3(u) cos(u) + constant
Substitute back for u = sin^(-1)(u/a):
= 1/8 a^4 sin^(-1)(u/a) + 1/4 u^3 sqrt(a^2 - u^2) - 1/8 a^3 u sqrt(1 - u^2/a^2) + constant
Factor the answer a different way:
= 1/8 (a^4 sin^(-1)(u/a) + 2 u^3 sqrt(a^2 - u^2) - a^3 u sqrt(1 - u^2/a^2)) + constant
Which is equivalent for restricted u and a values to:
Answer: |= 1/8 (u sqrt(a^2 - u^2) (2 u^2 - a^2) + a^4 tan^(-1)(u/sqrt(a^2 - u^2))) + constant
