求∫根号下（2ax-x²）dx

求∫根号下（2ax-x²）dx
求∫根号下（2ax-x²）dx

求∫根号下（2ax-x²）dx
∫ √(2ax - x²) dx
= ∫ √[a² - (x - a)²] dx
令x - a = asinz,dx = acosz dz
= ∫ |acosz|(acosz) dz
= a²∫ cos²z dz
= (a²/2)∫ (1 + cos2z) dz
= (a²/2)z + (a²/2)sinzcosz + C
= (a²/2)arcsin[(x - a)/a] + (a²/2)[(x - a)/a][√(2ax - x²)/a] + C
= (a²/2)arcsin(x/a - 1) + (1/2)(x - a)√(2ax - x²) + C