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

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

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integral of 1+2cos(θ)+cos^2(θ)

\int 1 + 2 cos (θ) + cos^{2} (θ) d θ

\frac{3}{2} θ + \frac{1}{4} sin (2 θ) + 2 sin (θ) + C

Solution steps

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

\int 1 d θ = θ

\int 2 cos (θ) d θ = 2 sin (θ)

\int cos^{2} (θ) d θ = \frac{1}{2} (θ + \frac{1}{2} sin (2 θ))

= θ + 2 sin (θ) + \frac{1}{2} (θ + \frac{1}{2} sin (2 θ))

Simplify

θ + 2 sin (θ) + \frac{1}{2} (θ + \frac{1}{2} sin (2 θ)) : \frac{3}{2} θ + \frac{1}{4} sin (2 θ) + 2 sin (θ)

= \frac{3}{2} θ + \frac{1}{4} sin (2 θ) + 2 sin (θ)

Add a constant to the solution

= \frac{3}{2} θ + \frac{1}{4} sin (2 θ) + 2 sin (θ) + C

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