What is the integral of ((x^4-x^5))/(x^5) ?

The integral of ((x^4-x^5))/(x^5) is ln|x|-x+C

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integral of ((x^4-x^5))/(x^5)

\int \frac{( x ^{4} - x ^{5} )}{x ^{5}} d x

ln ∣ x ∣ - x + C

Solution steps

\int \frac{x ^{4} - x ^{5}}{x ^{5}} d x

Expand

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

= \int \frac{1}{x} - 1 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} d x - \int 1 d x

\int \frac{1}{x} d x = ln ∣ x ∣

\int 1 d x = x

= ln ∣ x ∣ - x

Add a constant to the solution

= ln ∣ x ∣ - x + C

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What is the integral of ((x^4-x^5))/(x^5) ?
The integral of ((x^4-x^5))/(x^5) is ln|x|-x+C