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

The integral of (2+2cos(θ))^2 is 6θ+8sin(θ)+sin(2θ)+C

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integral of (2+2cos(θ))^2

\int (2 + 2 cos (θ))^{2} d θ

6 θ + 8 sin (θ) + sin (2 θ) + C

Solution steps

\int (2 + 2 cos (θ))^{2} d θ

Expand

(2 + 2 cos (θ))^{2} : 4 + 8 cos (θ) + 4 cos^{2} (θ)

= \int 4 + 8 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 4 d θ + \int 8 cos (θ) d θ + \int 4 cos^{2} (θ) d θ

\int 4 d θ = 4 θ

\int 8 cos (θ) d θ = 8 sin (θ)

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

= 4 θ + 8 sin (θ) + 2 θ + sin (2 θ)

Simplify

= 6 θ + 8 sin (θ) + sin (2 θ)

Add a constant to the solution

= 6 θ + 8 sin (θ) + sin (2 θ) + C

\int \frac{1}{4 x ^{2} + 4 x + 10} d x

\frac{d y}{d x} = (3 x^{2} y)

4 x, 8 x^{2}

f (x) = x^{2} - 4, at x = - 5, 21

\int \frac{x}{3 x ^{2} + 1} d x

What is the integral of (2+2cos(θ))^2 ?
The integral of (2+2cos(θ))^2 is 6θ+8sin(θ)+sin(2θ)+C