What is the integral of (z^5)/(\sqrt[3]{9+z^6)} ?

The integral of (z^5)/(\sqrt[3]{9+z^6)} is 1/4 (9+z^6)^{2/3}+C

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integral of (z^5)/(\sqrt[3]{9+z^6)}

\int \frac{z ^{5}}{3 9 + z ^{6}} d z

\frac{1}{4} (9 + z^{6})^{\frac{2}{3}} + C

Solution steps

\int \frac{z ^{5}}{3 9 + z ^{6}} d z

Apply u-substitution

: \int \frac{1}{6 3 u} d u

= \int \frac{1}{6 3 u} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int \frac{1}{3 u} d u

Apply the Power Rule

: \frac{3}{2} u^{\frac{2}{3}}

= \frac{1}{6} \cdot \frac{3}{2} u^{\frac{2}{3}}

Substitute back

u = 9 + z^{6}

= \frac{1}{6} \cdot \frac{3}{2} (9 + z^{6})^{\frac{2}{3}}

Simplify

\frac{1}{6} \cdot \frac{3}{2} (9 + z^{6})^{\frac{2}{3}} : \frac{1}{4} (9 + z^{6})^{\frac{2}{3}}

= \frac{1}{4} (9 + z^{6})^{\frac{2}{3}}

Add a constant to the solution

= \frac{1}{4} (9 + z^{6})^{\frac{2}{3}} + C

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What is the integral of (z^5)/(\sqrt[3]{9+z^6)} ?
The integral of (z^5)/(\sqrt[3]{9+z^6)} is 1/4 (9+z^6)^{2/3}+C