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

The integral of sin(2x)(cos(2x))^4 is -1/10 cos^5(2x)+C

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

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

- \frac{1}{10} cos^{5} (2 x) + C

Solution steps

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

Simplify

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

Apply u-substitution

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

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

Take the constant out:

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

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

Apply u-substitution

: \int - v^{4} d v

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

Take the constant out:

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

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

Apply the Power Rule

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

= \frac{1}{2} (- \frac{v ^{5}}{5})

Substitute back

= \frac{1}{2} (- \frac{cos ^{5} ( 2 x )}{5})

Simplify

\frac{1}{2} (- \frac{cos ^{5} ( 2 x )}{5}) : - \frac{1}{10} cos^{5} (2 x)

= - \frac{1}{10} cos^{5} (2 x)

Add a constant to the solution

= - \frac{1}{10} cos^{5} (2 x) + C

\int \frac{( 2 x - 3 )}{x ^{2} + 4 x + 3} d x

\int \frac{\frac{1}{2}}{x + 1} d x

\int 0.01 e^{- 0.01 x} d x

\int \frac{5 x}{8} d x

\int sin (x) - 4 d x

What is the integral of sin(2x)(cos(2x))^4 ?
The integral of sin(2x)(cos(2x))^4 is -1/10 cos^5(2x)+C