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

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

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integral of (x+5)/(\sqrt[3]{x+9)}

\int \frac{x + 5}{3 x + 9} d x

\frac{3}{5} (x + 9)^{\frac{5}{3}} - 6 (x + 9)^{\frac{2}{3}} + C

Solution steps

\int \frac{x + 5}{3 x + 9} d x

Expand

\frac{x + 5}{3 x + 9} : \frac{x}{3 x + 9} + \frac{5}{3 x + 9}

= \int \frac{x}{3 x + 9} + \frac{5}{3 x + 9} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x}{3 x + 9} d x + \int \frac{5}{3 x + 9} d x

\int \frac{x}{3 x + 9} d x = \frac{3}{5} (x + 9)^{\frac{5}{3}} - \frac{27}{2} (x + 9)^{\frac{2}{3}}

\int \frac{5}{3 x + 9} d x = \frac{15}{2} (x + 9)^{\frac{2}{3}}

= \frac{3}{5} (x + 9)^{\frac{5}{3}} - \frac{27}{2} (x + 9)^{\frac{2}{3}} + \frac{15}{2} (x + 9)^{\frac{2}{3}}

Add similar elements:

- \frac{27}{2} (x + 9)^{\frac{2}{3}} + \frac{15}{2} (x + 9)^{\frac{2}{3}} = - 6 (x + 9)^{\frac{2}{3}}

= \frac{3}{5} (x + 9)^{\frac{5}{3}} - 6 (x + 9)^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{5} (x + 9)^{\frac{5}{3}} - 6 (x + 9)^{\frac{2}{3}} + C

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