What is the integral of 8sin^3(θ)cos(θ) ?

The integral of 8sin^3(θ)cos(θ) is 2sin^4(θ)+C

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integral of 8sin^3(θ)cos(θ)

\int 8 sin^{3} (θ) cos (θ) d θ

2 sin^{4} (θ) + C

Solution steps

\int 8 sin^{3} (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^{3} (θ) cos (θ) d θ

Apply u-substitution

: \int u^{3} d u

= 8 \cdot \int u^{3} d u

Apply the Power Rule

: \frac{u ^{4}}{4}

= 8 \cdot \frac{u ^{4}}{4}

Substitute back

u = sin (θ)

= 8 \cdot \frac{sin ^{4} ( θ )}{4}

Simplify

8 \cdot \frac{sin ^{4} ( θ )}{4} : 2 sin^{4} (θ)

= 2 sin^{4} (θ)

Add a constant to the solution

= 2 sin^{4} (θ) + C

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What is the integral of 8sin^3(θ)cos(θ) ?
The integral of 8sin^3(θ)cos(θ) is 2sin^4(θ)+C