What is the integral of ((x^{-1/3}-5))/x ?

The integral of ((x^{-1/3}-5))/x is -3/(x^{1/3)}-5ln|x|+C

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integral of ((x^{-1/3}-5))/x

\int \frac{( x ^{- \frac{1}{3}} - 5 )}{x} d x

- \frac{3}{x ^{\frac{1}{3}}} - 5 ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{- \frac{1}{3}} - 5}{x} d x

Expand

\frac{x ^{- \frac{1}{3}} - 5}{x} : \frac{1}{x ^{\frac{4}{3}}} - \frac{5}{x}

= \int \frac{1}{x ^{\frac{4}{3}}} - \frac{5}{x} 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{1}{x ^{\frac{4}{3}}} d x - \int \frac{5}{x} d x

\int \frac{1}{x ^{\frac{4}{3}}} d x = - \frac{3}{x ^{\frac{1}{3}}}

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= - \frac{3}{x ^{\frac{1}{3}}} - 5 ln ∣ x ∣

Add a constant to the solution

= - \frac{3}{x ^{\frac{1}{3}}} - 5 ln ∣ x ∣ + C

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What is the integral of ((x^{-1/3}-5))/x ?
The integral of ((x^{-1/3}-5))/x is -3/(x^{1/3)}-5ln|x|+C