What is the integral of ax^{-1} ?

The integral of ax^{-1} is aln|x|+C

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integral of ax^{-1}

\int a x^{- 1} d x

a ln ∣ x ∣ + C

Solution steps

\int a x^{- 1} d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= a \cdot \int x^{- 1} d x

Use the common integral:

\int x^{- 1} d x = ln (∣ x ∣)

= a ln ∣ x ∣

Add a constant to the solution

= a ln ∣ x ∣ + C