What is the integral of ln(x^{-2}) ?

The integral of ln(x^{-2}) is -2xln(x)+2x+C

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integral of ln(x^{-2})

\int ln (x^{- 2}) d x

- 2 x ln (x) + 2 x + C

Solution steps

\int ln (x^{- 2}) d x

Simplify

ln (x^{- 2}) : - ln (x^{2})

= \int - ln (x^{2}) d x

Take the constant out:

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

= - \int ln (x^{2}) d x

Apply Integration By Parts

= - (x ln (x^{2}) - \int 2 d x)

\int 2 d x = 2 x

= - (x ln (x^{2}) - 2 x)

Simplify

- (x ln (x^{2}) - 2 x) : - 2 x ln (x) + 2 x

= - 2 x ln (x) + 2 x

Add a constant to the solution

= - 2 x ln (x) + 2 x + C

What is the integral of ln(x^{-2}) ?
The integral of ln(x^{-2}) is -2xln(x)+2x+C