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

The integral of sin^4(4x)cos(4x) is 1/20 sin^5(4x)+C

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integral of sin^4(4x)cos(4x)

\int sin^{4} (4 x) cos (4 x) d x

\frac{1}{20} sin^{5} (4 x) + C

Solution steps

\int sin^{4} (4 x) cos (4 x) d x

Apply u-substitution

: \int sin^{4} (u) cos (u) \frac{1}{4} d u

= \int sin^{4} (u) cos (u) \frac{1}{4} d u

Take the constant out:

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

= \frac{1}{4} \cdot \int sin^{4} (u) cos (u) d u

Apply u-substitution

: \int v^{4} d v

= \frac{1}{4} \cdot \int v^{4} d v

Apply the Power Rule

: \frac{v ^{5}}{5}

= \frac{1}{4} \cdot \frac{v ^{5}}{5}

Substitute back

= \frac{1}{4} \cdot \frac{sin ^{5} ( 4 x )}{5}

Simplify

\frac{1}{4} \cdot \frac{sin ^{5} ( 4 x )}{5} : \frac{1}{20} sin^{5} (4 x)

= \frac{1}{20} sin^{5} (4 x)

Add a constant to the solution

= \frac{1}{20} sin^{5} (4 x) + C

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What is the integral of sin^4(4x)cos(4x) ?
The integral of sin^4(4x)cos(4x) is 1/20 sin^5(4x)+C