What is the integral of (1+sqrt(cos(x)))^2sin(x) ?

The integral of (1+sqrt(cos(x)))^2sin(x) is -cos(x)-4/3 cos^{3/2}(x)+(sin^2(x))/2+C

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

\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

Solution steps

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

Expand

(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

Apply the Sum Rule:

\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}

Add a constant to the solution

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

What is the integral of (1+sqrt(cos(x)))^2sin(x) ?
The integral of (1+sqrt(cos(x)))^2sin(x) is -cos(x)-4/3 cos^{3/2}(x)+(sin^2(x))/2+C