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

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

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

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

2 (- 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 3 ∣) + C

Solution steps

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

Take the partial fraction of

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

= 2 \cdot \int - \frac{2}{x + 2} + \frac{3}{x + 3} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

= 2 (- 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 3 ∣)

Add a constant to the solution

= 2 (- 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 3 ∣) + C

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