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

The integral of (x+3)/(\sqrt[3]{x-5)} is 3/5 (x-5)^{5/3}+12(x-5)^{2/3}+C

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

\frac{3}{5} (x - 5)^{\frac{5}{3}} + 12 (x - 5)^{\frac{2}{3}} + C

Solution steps

Expand

Apply the Sum Rule:

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

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

Add similar elements:

\frac{15}{2} (x - 5)^{\frac{2}{3}} + \frac{9}{2} (x - 5)^{\frac{2}{3}} = 12 (x - 5)^{\frac{2}{3}}

= \frac{3}{5} (x - 5)^{\frac{5}{3}} + 12 (x - 5)^{\frac{2}{3}}

Add a constant to the solution

= \frac{3}{5} (x - 5)^{\frac{5}{3}} + 12 (x - 5)^{\frac{2}{3}} + C

What is the integral of (x+3)/(\sqrt[3]{x-5)} ?
The integral of (x+3)/(\sqrt[3]{x-5)} is 3/5 (x-5)^{5/3}+12(x-5)^{2/3}+C