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

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

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

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

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

Solution steps

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

Expand

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

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

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

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

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

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

Add a constant to the solution

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

\int (arctan (t))^{2} d t

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\int e ln (x) d x

\int \frac{x ^{2} ln ( x )}{( x ^{3} + 1 ) ^{2}} d x

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

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