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

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

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integral of \sqrt[3]{1+3sin(x)}cos(x)

\int 3 1 + 3 sin (x) cos (x) d x

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

Solution steps

\int 3 1 + 3 sin (x) cos (x) d x

Apply u-substitution

: \int 3 1 + 3 u d u

= \int 3 1 + 3 u d u

Apply u-substitution

: \int \frac{3 v}{3} d v

= \int \frac{3 v}{3} d v

Take the constant out:

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

= \frac{1}{3} \cdot \int 3 v d v

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

Simplify

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

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

Add a constant to the solution

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

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