What is the integral of ln(x^{-1}) ?

The integral of ln(x^{-1}) is -xln(x)+x+C

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integral of ln(x^{-1})

\int ln (x^{- 1}) d x

- x ln (x) + x + C

Solution steps

\int ln (x^{- 1}) d x

Simplify

ln (x^{- 1}) : - ln (x)

= \int - ln (x) d x

Take the constant out:

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

= - \int ln (x) d x

Apply Integration By Parts

: x ln (x) - \int 1 d x

= - (x ln (x) - \int 1 d x)

\int 1 d x = x

= - (x ln (x) - x)

Simplify

= - x ln (x) + x

Add a constant to the solution

= - x ln (x) + x + C

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