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

The integral of sin^2(3θ)cos^2(3θ) is 1/8 (θ-1/12 sin(12θ))+C

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integral of sin^2(3θ)cos^2(3θ)

\int sin^{2} (3 θ) cos^{2} (3 θ) d θ

\frac{1}{8} (θ - \frac{1}{12} sin (12 θ)) + C

Solution steps

\int sin^{2} (3 θ) cos^{2} (3 θ) d θ

Rewrite using trig identities

= \int \frac{1}{8} (1 - cos (12 θ)) d θ

Take the constant out:

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

= \frac{1}{8} \cdot \int 1 - cos (12 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}{8} (\int 1 d θ - \int cos (12 θ) d θ)

\int 1 d θ = θ

\int cos (12 θ) d θ = \frac{1}{12} sin (12 θ)

= \frac{1}{8} (θ - \frac{1}{12} sin (12 θ))

Add a constant to the solution

= \frac{1}{8} (θ - \frac{1}{12} sin (12 θ)) + C

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What is the integral of sin^2(3θ)cos^2(3θ) ?
The integral of sin^2(3θ)cos^2(3θ) is 1/8 (θ-1/12 sin(12θ))+C