What is the integral of x/((5x-4)^2) ?

The integral of x/((5x-4)^2) is -x/(5(5x-4))+1/25 ln|5x-4|+C

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integral of x/((5x-4)^2)

\int \frac{x}{( 5 x - 4 ) ^{2}} d x

- \frac{x}{5 ( 5 x - 4 )} + \frac{1}{25} ln ∣ 5 x - 4 ∣ + C

Solution steps

\int \frac{x}{( 5 x - 4 ) ^{2}} d x

Apply Integration By Parts

: - \frac{x}{5 ( 5 x - 4 )} - \int - \frac{1}{5 ( 5 x - 4 )} d x

= - \frac{x}{5 ( 5 x - 4 )} - \int - \frac{1}{5 ( 5 x - 4 )} d x

\int - \frac{1}{5 ( 5 x - 4 )} d x = - \frac{1}{25} ln ∣ 5 x - 4 ∣

= - \frac{x}{5 ( 5 x - 4 )} - (- \frac{1}{25} ln ∣ 5 x - 4 ∣)

Simplify

= - \frac{x}{5 ( 5 x - 4 )} + \frac{1}{25} ln ∣ 5 x - 4 ∣

Add a constant to the solution

= - \frac{x}{5 ( 5 x - 4 )} + \frac{1}{25} ln ∣ 5 x - 4 ∣ + C

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What is the integral of x/((5x-4)^2) ?
The integral of x/((5x-4)^2) is -x/(5(5x-4))+1/25 ln|5x-4|+C