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

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

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

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

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

Solution steps

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

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

Simplify

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

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

Add a constant to the solution

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

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

4 (8 x - 1)^{4}

x^{^{''}} + 4 x^{^{'}} + 4 x = e^{- 2 t}

f (x) = 3 x^{3}

\int_{0}^{8} - x^{2} + 2 x + 4 d x

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