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

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

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integral of (3+2cos(θ))^2

\int (3 + 2 cos (θ))^{2} d θ

11 θ + 12 sin (θ) + sin (2 θ) + C

Solution steps

\int (3 + 2 cos (θ))^{2} d θ

Expand

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

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

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 θ + \int 12 cos (θ) d θ + \int 4 cos^{2} (θ) d θ

\int 9 d θ = 9 θ

\int 12 cos (θ) d θ = 12 sin (θ)

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

= 9 θ + 12 sin (θ) + 2 θ + sin (2 θ)

Simplify

= 11 θ + 12 sin (θ) + sin (2 θ)

Add a constant to the solution

= 11 θ + 12 sin (θ) + sin (2 θ) + C

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