What is the integral of cos^4(θ)sin(2θ) ?

The integral of cos^4(θ)sin(2θ) is -1/3 cos^6(θ)+C

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integral of cos^4(θ)sin(2θ)

\int cos^{4} (θ) sin (2 θ) d θ

- \frac{1}{3} cos^{6} (θ) + C

Solution steps

\int cos^{4} (θ) sin (2 θ) d θ

Rewrite using trig identities

= \int 2 cos^{5} (θ) sin (θ) d θ

Take the constant out:

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

= 2 \cdot \int cos^{5} (θ) sin (θ) d θ

Apply u-substitution

: \int - u^{5} d u

= 2 \cdot \int - u^{5} d u

Take the constant out:

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

= 2 (- \int u^{5} d u)

Apply the Power Rule

: \frac{u ^{6}}{6}

= 2 (- \frac{u ^{6}}{6})

Substitute back

u = cos (θ)

= 2 (- \frac{cos ^{6} ( θ )}{6})

Simplify

2 (- \frac{cos ^{6} ( θ )}{6}) : - \frac{1}{3} cos^{6} (θ)

= - \frac{1}{3} cos^{6} (θ)

Add a constant to the solution

= - \frac{1}{3} cos^{6} (θ) + C

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What is the integral of cos^4(θ)sin(2θ) ?
The integral of cos^4(θ)sin(2θ) is -1/3 cos^6(θ)+C