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

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

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

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

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

Solution steps

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

\frac{\partial}{\partial y} (x e^{x^{3}})

y = 2 x - 1

x \to 7 lim (\frac{π x}{6})

\int \frac{5 x}{( x - 2 ) ^{2}} d x

\frac{d}{d θ} (tan (\frac{θ}{2}))

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