What is the integral of ((-2+ln(-2x))^{-3})/x ?

The integral of ((-2+ln(-2x))^{-3})/x is -1/(2(-2+ln(-2x))^2)+C

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integral of ((-2+ln(-2x))^{-3})/x

\int \frac{( - 2 + ln ( - 2 x ) ) ^{- 3}}{x} d x

- \frac{1}{2 ( - 2 + ln ( - 2 x ) ) ^{2}} + C

Solution steps

\int \frac{( - 2 + ln ( - 2 x ) ) ^{- 3}}{x} d x

Apply u-substitution

: \int \frac{1}{u ^{3}} d u

= \int \frac{1}{u ^{3}} d u

Apply the Power Rule

: - \frac{1}{2 u ^{2}}

= - \frac{1}{2 u ^{2}}

Substitute back

u = - 2 + ln (- 2 x)

= - \frac{1}{2 ( - 2 + ln ( - 2 x ) ) ^{2}}

Add a constant to the solution

= - \frac{1}{2 ( - 2 + ln ( - 2 x ) ) ^{2}} + C

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What is the integral of ((-2+ln(-2x))^{-3})/x ?
The integral of ((-2+ln(-2x))^{-3})/x is -1/(2(-2+ln(-2x))^2)+C