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

The integral of (x+5)/(\sqrt[5]{x-5)} is 5/9 (x-5)^{9/5}+25/2 (x-5)^{4/5}+C

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

\frac{5}{9} (x - 5)^{\frac{9}{5}} + \frac{25}{2} (x - 5)^{\frac{4}{5}} + C

Solution steps

Apply u-substitution

Expand

Apply the Sum Rule:

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

\int u^{\frac{4}{5}} d u = \frac{5}{9} u^{\frac{9}{5}}

= \frac{5}{9} u^{\frac{9}{5}} + \frac{25}{2} u^{\frac{4}{5}}

Substitute back

u = x - 5

= \frac{5}{9} (x - 5)^{\frac{9}{5}} + \frac{25}{2} (x - 5)^{\frac{4}{5}}

Add a constant to the solution

= \frac{5}{9} (x - 5)^{\frac{9}{5}} + \frac{25}{2} (x - 5)^{\frac{4}{5}} + C

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What is the integral of (x+5)/(\sqrt[5]{x-5)} ?
The integral of (x+5)/(\sqrt[5]{x-5)} is 5/9 (x-5)^{9/5}+25/2 (x-5)^{4/5}+C