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

The integral of x/((2+3x)^2) is -x/(3(2+3x))+1/9 ln|2+3x|+C

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

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

- \frac{x}{3 ( 2 + 3 x )} + \frac{1}{9} ln ∣ 2 + 3 x ∣ + C

Solution steps

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

Apply Integration By Parts

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

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

\int - \frac{1}{3 ( 2 + 3 x )} d x = - \frac{1}{9} ln ∣ 2 + 3 x ∣

= - \frac{x}{3 ( 2 + 3 x )} - (- \frac{1}{9} ln ∣ 2 + 3 x ∣)

Simplify

= - \frac{x}{3 ( 2 + 3 x )} + \frac{1}{9} ln ∣ 2 + 3 x ∣

Add a constant to the solution

= - \frac{x}{3 ( 2 + 3 x )} + \frac{1}{9} ln ∣ 2 + 3 x ∣ + C

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What is the integral of x/((2+3x)^2) ?
The integral of x/((2+3x)^2) is -x/(3(2+3x))+1/9 ln|2+3x|+C