What is the integral of cos^3(7x)sin(7x) ?

The integral of cos^3(7x)sin(7x) is -1/28 cos^4(7x)+C

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integral of cos^3(7x)sin(7x)

\int cos^{3} (7 x) sin (7 x) d x

- \frac{1}{28} cos^{4} (7 x) + C

Solution steps

\int cos^{3} (7 x) sin (7 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}{21} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = cos^{3} (7 x)

= - \frac{1}{21} \cdot \frac{3}{4} (cos^{3} (7 x))^{\frac{4}{3}}

Simplify

- \frac{1}{21} \cdot \frac{3}{4} (cos^{3} (7 x))^{\frac{4}{3}} : - \frac{1}{28} cos^{4} (7 x)

= - \frac{1}{28} cos^{4} (7 x)

Add a constant to the solution

= - \frac{1}{28} cos^{4} (7 x) + C

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What is the integral of cos^3(7x)sin(7x) ?
The integral of cos^3(7x)sin(7x) is -1/28 cos^4(7x)+C