What is the integral of x/((2x+3)^2) ?

The integral of x/((2x+3)^2) is -x/(2(2x+3))+1/4 ln|2x+3|+C

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integral of x/((2x+3)^2)

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

- \frac{x}{2 ( 2 x + 3 )} + \frac{1}{4} ln ∣ 2 x + 3 ∣ + C

Solution steps

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

Apply Integration By Parts

: - \frac{x}{2 ( 2 x + 3 )} - \int - \frac{1}{2 ( 2 x + 3 )} d x

= - \frac{x}{2 ( 2 x + 3 )} - \int - \frac{1}{2 ( 2 x + 3 )} d x

\int - \frac{1}{2 ( 2 x + 3 )} d x = - \frac{1}{4} ln ∣ 2 x + 3 ∣

= - \frac{x}{2 ( 2 x + 3 )} - (- \frac{1}{4} ln ∣ 2 x + 3 ∣)

Simplify

= - \frac{x}{2 ( 2 x + 3 )} + \frac{1}{4} ln ∣ 2 x + 3 ∣

Add a constant to the solution

= - \frac{x}{2 ( 2 x + 3 )} + \frac{1}{4} ln ∣ 2 x + 3 ∣ + C

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What is the integral of x/((2x+3)^2) ?
The integral of x/((2x+3)^2) is -x/(2(2x+3))+1/4 ln|2x+3|+C