What is the integral of cos^5(4x)sin(4x) ?

The integral of cos^5(4x)sin(4x) is -1/24 cos^6(4x)+C

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integral of cos^5(4x)sin(4x)

\int cos^{5} (4 x) sin (4 x) d x

- \frac{1}{24} cos^{6} (4 x) + C

Solution steps

\int cos^{5} (4 x) sin (4 x) d x

Apply u-substitution

Take the constant out:

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

Apply the Power Rule

: \frac{5}{6} u^{\frac{6}{5}}

= - \frac{1}{20} \cdot \frac{5}{6} u^{\frac{6}{5}}

Substitute back

u = cos^{5} (4 x)

= - \frac{1}{20} \cdot \frac{5}{6} (cos^{5} (4 x))^{\frac{6}{5}}

Simplify

- \frac{1}{20} \cdot \frac{5}{6} (cos^{5} (4 x))^{\frac{6}{5}} : - \frac{1}{24} cos^{6} (4 x)

= - \frac{1}{24} cos^{6} (4 x)

Add a constant to the solution

= - \frac{1}{24} cos^{6} (4 x) + C

\frac{d}{d x} (ln (6 x^{2} + 2 x + 2))

\frac{d}{d t} (e^{\frac{t}{7}} cos (\frac{t}{7}))

\frac{d}{d x} (\frac{2 ^{x}}{e ^{x}})

y^{^{'}} = y (y - 1)

f (x) = \frac{4 x + 3}{x + 1}, at x = 3

What is the integral of cos^5(4x)sin(4x) ?
The integral of cos^5(4x)sin(4x) is -1/24 cos^6(4x)+C