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

The integral of sin(4x)cos^2(4x) is -1/12 cos^3(4x)+C

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integral of sin(4x)cos^2(4x)

\int sin (4 x) cos^{2} (4 x) d x

- \frac{1}{12} cos^{3} (4 x) + C

Solution steps

\int sin (4 x) cos^{2} (4 x) d x

Apply u-substitution

: \int sin (u) cos^{2} (u) \frac{1}{4} d u

= \int sin (u) cos^{2} (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 (u) cos^{2} (u) d u

Apply u-substitution

: \int - v^{2} d v

= \frac{1}{4} \cdot \int - v^{2} d v

Take the constant out:

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

= \frac{1}{4} (- \int v^{2} d v)

Apply the Power Rule

: \frac{v ^{3}}{3}

= \frac{1}{4} (- \frac{v ^{3}}{3})

Substitute back

= \frac{1}{4} (- \frac{cos ^{3} ( 4 x )}{3})

Simplify

\frac{1}{4} (- \frac{cos ^{3} ( 4 x )}{3}) : - \frac{1}{12} cos^{3} (4 x)

= - \frac{1}{12} cos^{3} (4 x)

Add a constant to the solution

= - \frac{1}{12} cos^{3} (4 x) + C

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What is the integral of sin(4x)cos^2(4x) ?
The integral of sin(4x)cos^2(4x) is -1/12 cos^3(4x)+C