What is the integral from 0 to 13 of x/((x+2)(x+3)) ?

The integral from 0 to 13 of x/((x+2)(x+3)) is -2ln(15)+14ln(2)-3ln(3)

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integral from 0 to 13 of x/((x+2)(x+3))

\int_{0}^{13} \frac{x}{( x + 2 ) ( x + 3 )} d x

- 2 ln (15) + 14 ln (2) - 3 ln (3)

Decimal Notation

0.99212 \dots

Solution steps

\int_{0}^{13} \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_{0}^{13} - \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_{0}^{13} \frac{2}{x + 2} d x + \int_{0}^{13} \frac{3}{x + 3} d x

\int_{0}^{13} \frac{2}{x + 2} d x = 2 (ln (15) - ln (2))

\int_{0}^{13} \frac{3}{x + 3} d x = 3 (4 ln (2) - ln (3))

= - 2 (ln (15) - ln (2)) + 3 (4 ln (2) - ln (3))

Simplify

- 2 (ln (15) - ln (2)) + 3 (4 ln (2) - ln (3)) : - 2 ln (15) + 14 ln (2) - 3 ln (3)

= - 2 ln (15) + 14 ln (2) - 3 ln (3)

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What is the integral from 0 to 13 of x/((x+2)(x+3)) ?
The integral from 0 to 13 of x/((x+2)(x+3)) is -2ln(15)+14ln(2)-3ln(3)