What is the integral of (cos(x))/((sin^2(x))^{1/3)} ?

The integral of (cos(x))/((sin^2(x))^{1/3)} is 3sin^{1/3}(x)+C

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integral of (cos(x))/((sin^2(x))^{1/3)}

\int \frac{cos ( x )}{( sin ^{2} ( x ) ) ^{\frac{1}{3}}} d x

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

Solution steps

\int \frac{cos ( x )}{( sin ^{2} ( x ) ) ^{\frac{1}{3}}} d x

(sin^{2} (x))^{\frac{1}{3}} = sin^{\frac{2}{3}} (x),

assuming

sin (x) \geq 0

= \int \frac{cos ( x )}{sin ^{\frac{2}{3}} ( x )} d x

Apply u-substitution

: \int \frac{1}{u ^{\frac{2}{3}}} d u

= \int \frac{1}{u ^{\frac{2}{3}}} d u

Apply the Power Rule

: 3 u^{\frac{1}{3}}

= 3 u^{\frac{1}{3}}

Substitute back

u = sin (x)

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

Add a constant to the solution

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

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What is the integral of (cos(x))/((sin^2(x))^{1/3)} ?
The integral of (cos(x))/((sin^2(x))^{1/3)} is 3sin^{1/3}(x)+C