integral of (1+sqrt(cos(x)))^2sin(x)

解答

\int (1 + cos (x))^{2} sin (x) d x

- cos (x) - \frac{4}{3} cos^{\frac{3}{2}} (x) + \frac{sin ^{2} ( x )}{2} + C

求解步骤

\int (1 + cos (x))^{2} sin (x) d x

乘开

(1 + cos (x))^{2} sin (x) : sin (x) + 2 sin (x) cos (x) + sin (x) cos (x)

= \int sin (x) + 2 sin (x) cos (x) + sin (x) cos (x) d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int sin (x) d x + \int 2 sin (x) cos (x) d x + \int sin (x) cos (x) d x

\int sin (x) d x = - cos (x)

\int 2 sin (x) cos (x) d x = - \frac{4}{3} cos^{\frac{3}{2}} (x)

\int sin (x) cos (x) d x = \frac{sin ^{2} ( x )}{2}

= - cos (x) - \frac{4}{3} cos^{\frac{3}{2}} (x) + \frac{sin ^{2} ( x )}{2}

解答补常数

= - cos (x) - \frac{4}{3} cos^{\frac{3}{2}} (x) + \frac{sin ^{2} ( x )}{2} + C