What is the integral of cos^2(2θ) ?

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

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integral of cos^2(2θ)

\int cos^{2} (2 θ) d θ

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

Solution steps

\int cos^{2} (2 θ) d θ

Rewrite using trig identities

= \int \frac{1}{2} (1 + cos (4 θ)) d θ

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int 1 + cos (4 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

= \frac{1}{2} (\int 1 d θ + \int cos (4 θ) d θ)

\int 1 d θ = θ

\int cos (4 θ) d θ = \frac{1}{4} sin (4 θ)

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

Add a constant to the solution

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

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

\frac{\partial}{\partial x} (2 x^{2} + 9 y^{2})

\int 2 e^{3 x + e^{3 x}} d x

\frac{d}{d t} (2 e^{t} cos (t))

\frac{d P}{d t} + 2 tP = P + 6 t - 3

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