Evaluate \[ \int(\sin x+\cos x)^2dx, \] \[ \int\frac32\sin(2x)(\sin x+\cos x)dx, \quad \int\cos^2x(1+\sin x)dx, \] \[ \int\sin^3x\cos^5x\,dx, \quad \int\cos^3x\sin x\,dx, \quad \int\sin^3x\sqrt{\cos x}\,dx. \]
Solution
Direct expansion/substitution gives, respectively, \[ \boxed{x-\frac12\cos2x+C,} \] \[ \boxed{\sin^3x-\cos^3x+C,} \] \[ \boxed{-\frac13\cos^3x+\frac12(x+\sin x\cos x)+C,} \] \[ \boxed{-\frac16\cos^6x+\frac18\cos^8x+C,} \] \[ \boxed{-\frac14\cos^4x+C,} \] and, on intervals where \(\sqrt{\cos x}\) is real, \[ \boxed{-\frac23\cos^{3/2}x+\frac27\cos^{7/2}x+C.} \] For example, the fourth follows by writing \(\sin^3x=\sin x(1-\cos^2x)\) and taking \(u=\cos x\); the other odd-power cases are analogous.