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

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

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

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

- \frac{1}{3} cos^{6} (x) + C

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

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

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 (x)

= 2 (- \frac{cos ^{6} ( x )}{6})

Simplify

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

= - \frac{1}{3} cos^{6} (x)

Add a constant to the solution

= - \frac{1}{3} cos^{6} (x) + C

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