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

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

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

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

- 2 ln ∣ x + 2 ∣ + 3 ln ∣ x + 3 ∣ + C

Solution steps

\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}

= \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

= - \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 ln ∣ x + 2 ∣ + 3 ln ∣ x + 3 ∣

Add a constant to the solution

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

\frac{1}{s ^{2}} - \frac{48}{s ^{5}}

\frac{d}{d x} (\frac{x ^{3} - 6}{x ^{2}})

x = \frac{( y - 2 ) ^{2}}{3}, y = 8 - x

\int \frac{1}{x ^{\frac{1}{2}} + x ^{\frac{1}{3}}} d x

\int 4 x^{2} - x + 1 d x

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