What is the integral of 5/(x(-1+ln(-2x))^3) ?

The integral of 5/(x(-1+ln(-2x))^3) is -5/(2(-1+ln(-2x))^2)+C

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integral of 5/(x(-1+ln(-2x))^3)

\int \frac{5}{x ( - 1 + ln ( - 2 x ) ) ^{3}} d x

- \frac{5}{2 ( - 1 + ln ( - 2 x ) ) ^{2}} + C

Solution steps

\int \frac{5}{x ( - 1 + ln ( - 2 x ) ) ^{3}} d x

Take the constant out:

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

= 5 \cdot \int \frac{1}{x ( - 1 + ln ( - 2 x ) ) ^{3}} d x

Apply u-substitution

: \int \frac{1}{u ( - 1 + ln ( u ) ) ^{3}} d u

= 5 \cdot \int \frac{1}{u ( - 1 + ln ( u ) ) ^{3}} d u

Apply u-substitution

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

= 5 \cdot \int \frac{1}{v ^{3}} d v

Apply the Power Rule

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

= 5 (- \frac{1}{2 v ^{2}})

Substitute back

= 5 (- \frac{1}{2 ( - 1 + ln ( - 2 x ) ) ^{2}})

Simplify

5 (- \frac{1}{2 ( - 1 + ln ( - 2 x ) ) ^{2}}) : - \frac{5}{2 ( - 1 + ln ( - 2 x ) ) ^{2}}

= - \frac{5}{2 ( - 1 + ln ( - 2 x ) ) ^{2}}

Add a constant to the solution

= - \frac{5}{2 ( - 1 + ln ( - 2 x ) ) ^{2}} + C

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What is the integral of 5/(x(-1+ln(-2x))^3) ?
The integral of 5/(x(-1+ln(-2x))^3) is -5/(2(-1+ln(-2x))^2)+C