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

The integral of (1+2cos(x))^2 is 3x+4sin(x)+sin(2x)+C

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

integral

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

3 x + 4 sin (x) + sin (2 x) + C

Solution steps

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

Expand

(1 + 2 cos (x))^{2} : 1 + 4 cos (x) + 4 cos^{2} (x)

= \int 1 + 4 cos (x) + 4 cos^{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 1 d x + \int 4 cos (x) d x + \int 4 cos^{2} (x) d x

\int 1 d x = x

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

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

= x + 4 sin (x) + 2 x + sin (2 x)

Simplify

= 3 x + 4 sin (x) + sin (2 x)

Add a constant to the solution

= 3 x + 4 sin (x) + sin (2 x) + C

What is the integral of (1+2cos(x))^2 ?
The integral of (1+2cos(x))^2 is 3x+4sin(x)+sin(2x)+C