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

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

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integral of x/((2x+5)^2)

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

- \frac{x}{2 ( 2 x + 5 )} + \frac{1}{4} ln ∣ 2 x + 5 ∣ + C

Solution steps

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

Apply Integration By Parts

: - \frac{x}{2 ( 2 x + 5 )} - \int - \frac{1}{2 ( 2 x + 5 )} d x

= - \frac{x}{2 ( 2 x + 5 )} - \int - \frac{1}{2 ( 2 x + 5 )} d x

\int - \frac{1}{2 ( 2 x + 5 )} d x = - \frac{1}{4} ln ∣ 2 x + 5 ∣

= - \frac{x}{2 ( 2 x + 5 )} - (- \frac{1}{4} ln ∣ 2 x + 5 ∣)

Simplify

= - \frac{x}{2 ( 2 x + 5 )} + \frac{1}{4} ln ∣ 2 x + 5 ∣

Add a constant to the solution

= - \frac{x}{2 ( 2 x + 5 )} + \frac{1}{4} ln ∣ 2 x + 5 ∣ + C

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