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

The integral of cos(x)(2+sin(x))^2 is 4sin(x)+2sin^2(x)+(sin^3(x))/3+C

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

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

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

Solution steps

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

Expand

cos (x) (2 + sin (x))^{2} : 4 cos (x) + 4 cos (x) sin (x) + sin^{2} (x) cos (x)

= \int 4 cos (x) + 4 cos (x) sin (x) + sin^{2} (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 4 cos (x) d x + \int 4 cos (x) sin (x) d x + \int sin^{2} (x) cos (x) d x

\int 4 cos (x) d x = 4 sin (x)

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

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

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

Add a constant to the solution

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

What is the integral of cos(x)(2+sin(x))^2 ?
The integral of cos(x)(2+sin(x))^2 is 4sin(x)+2sin^2(x)+(sin^3(x))/3+C