What is the integral of cos^3(4x)sin(4x) ?

The integral of cos^3(4x)sin(4x) is -1/16 cos^4(4x)+C

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integral of cos^3(4x)sin(4x)

\int cos^{3} (4 x) sin (4 x) d x

- \frac{1}{16} cos^{4} (4 x) + C

Solution steps

\int cos^{3} (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{3}{4} u^{\frac{4}{3}}

= - \frac{1}{12} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = cos^{3} (4 x)

= - \frac{1}{12} \cdot \frac{3}{4} (cos^{3} (4 x))^{\frac{4}{3}}

Simplify

- \frac{1}{12} \cdot \frac{3}{4} (cos^{3} (4 x))^{\frac{4}{3}} : - \frac{1}{16} cos^{4} (4 x)

= - \frac{1}{16} cos^{4} (4 x)

Add a constant to the solution

= - \frac{1}{16} cos^{4} (4 x) + C

\frac{d y}{d x} + tan (x) y - sec (x) = 0

\frac{d}{d x} (26 - sec^{3} (4 x))

x \to \infty lim ((4^{x} + 8^{x})^{\frac{1}{2 \cdot x}})

\frac{d}{d x} (- x cos (x) + x)

\frac{\partial}{\partial y} (x^{4} + y^{4} + 3 x^{2} y^{3})

What is the integral of cos^3(4x)sin(4x) ?
The integral of cos^3(4x)sin(4x) is -1/16 cos^4(4x)+C