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

The integral of (2+sin(x))^2 is 9/2 x-1/4 sin(2x)-4cos(x)+C

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

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

\frac{9}{2} x - \frac{1}{4} sin (2 x) - 4 cos (x) + C

Solution steps

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

Expand

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

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

\int 4 d x = 4 x

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

\int sin^{2} (x) d x = \frac{1}{2} (x - \frac{1}{2} sin (2 x))

= 4 x - 4 cos (x) + \frac{1}{2} (x - \frac{1}{2} sin (2 x))

Simplify

4 x - 4 cos (x) + \frac{1}{2} (x - \frac{1}{2} sin (2 x)) : \frac{9}{2} x - \frac{1}{4} sin (2 x) - 4 cos (x)

= \frac{9}{2} x - \frac{1}{4} sin (2 x) - 4 cos (x)

Add a constant to the solution

= \frac{9}{2} x - \frac{1}{4} sin (2 x) - 4 cos (x) + C

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What is the integral of (2+sin(x))^2 ?
The integral of (2+sin(x))^2 is 9/2 x-1/4 sin(2x)-4cos(x)+C