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

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

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

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

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

Add a constant to the solution

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

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