What is the integral of cos(θ)(1-cos(2θ))^3 ?

The integral of cos(θ)(1-cos(2θ))^3 is 8/7 sin^7(θ)+C

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integral of cos(θ)(1-cos(2θ))^3

\int cos (θ) (1 - cos (2 θ))^{3} d θ

\frac{8}{7} sin^{7} (θ) + C

Solution steps

\int cos (θ) (1 - cos (2 θ))^{3} d θ

Rewrite using trig identities

= \int 8 sin^{6} (θ) cos (θ) d θ

Take the constant out:

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

= 8 \cdot \int sin^{6} (θ) cos (θ) d θ

Apply u-substitution

: \int u^{6} d u

= 8 \cdot \int u^{6} d u

Apply the Power Rule

: \frac{u ^{7}}{7}

= 8 \cdot \frac{u ^{7}}{7}

Substitute back

u = sin (θ)

= 8 \cdot \frac{sin ^{7} ( θ )}{7}

Simplify

8 \cdot \frac{sin ^{7} ( θ )}{7} : \frac{8}{7} sin^{7} (θ)

= \frac{8}{7} sin^{7} (θ)

Add a constant to the solution

= \frac{8}{7} sin^{7} (θ) + C

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What is the integral of cos(θ)(1-cos(2θ))^3 ?
The integral of cos(θ)(1-cos(2θ))^3 is 8/7 sin^7(θ)+C