What is the integral of cos(3x)\sqrt[3]{sin(3x)} ?

The integral of cos(3x)\sqrt[3]{sin(3x)} is 1/4 sin^{4/3}(3x)+C

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integral of cos(3x)\sqrt[3]{sin(3x)}

\frac{1}{4} sin^{\frac{4}{3}} (3 x) + C

Solution steps

Apply u-substitution

Take the constant out:

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

Apply u-substitution

Apply the Power Rule

: \frac{3}{4} v^{\frac{4}{3}}

= \frac{1}{3} \cdot \frac{3}{4} v^{\frac{4}{3}}

Substitute back

= \frac{1}{3} \cdot \frac{3}{4} sin^{\frac{4}{3}} (3 x)

Simplify

\frac{1}{3} \cdot \frac{3}{4} sin^{\frac{4}{3}} (3 x) : \frac{1}{4} sin^{\frac{4}{3}} (3 x)

= \frac{1}{4} sin^{\frac{4}{3}} (3 x)

Add a constant to the solution

= \frac{1}{4} sin^{\frac{4}{3}} (3 x) + C

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What is the integral of cos(3x)\sqrt[3]{sin(3x)} ?
The integral of cos(3x)\sqrt[3]{sin(3x)} is 1/4 sin^{4/3}(3x)+C