What is the integral of (3+2cos(x))^2 ?

The integral of (3+2cos(x))^2 is 11x+12sin(x)+sin(2x)+C

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integral of (3+2cos(x))^2

\int (3 + 2 cos (x))^{2} d x

11 x + 12 sin (x) + sin (2 x) + C

Solution steps

\int (3 + 2 cos (x))^{2} d x

Expand

(3 + 2 cos (x))^{2} : 9 + 12 cos (x) + 4 cos^{2} (x)

= \int 9 + 12 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 9 d x + \int 12 cos (x) d x + \int 4 cos^{2} (x) d x

\int 9 d x = 9 x

\int 12 cos (x) d x = 12 sin (x)

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

= 9 x + 12 sin (x) + 2 x + sin (2 x)

Simplify

= 11 x + 12 sin (x) + sin (2 x)

Add a constant to the solution

= 11 x + 12 sin (x) + sin (2 x) + C

x \to 2 lim (\frac{6 x}{x - 2})

f (x) = 11 sin (x), at x = \frac{( 3 π )}{4}

\int_{1}^{4} \frac{( 2 x + 1 )}{2 x} d x

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

x \to 2 lim (\frac{1}{x ^{2} - 4 x + 4})

What is the integral of (3+2cos(x))^2 ?
The integral of (3+2cos(x))^2 is 11x+12sin(x)+sin(2x)+C