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

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

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integral of cos^2(θ)sin(2θ)

\int cos^{2} (θ) sin (2 θ) d θ

- \frac{1}{2} cos^{4} (θ) + C

Solution steps

\int cos^{2} (θ) sin (2 θ) d θ

Apply u-substitution

: \int - u d u

= \int - u d u

Take the constant out:

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

= - \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= - \frac{u ^{2}}{2}

Substitute back

u = cos^{2} (θ)

= - \frac{( cos ^{2} ( θ ) ) ^{2}}{2}

Simplify

- \frac{( cos ^{2} ( θ ) ) ^{2}}{2} : - \frac{1}{2} cos^{4} (θ)

= - \frac{1}{2} cos^{4} (θ)

Add a constant to the solution

= - \frac{1}{2} cos^{4} (θ) + C

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