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

解

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

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

十進法表記

0.99212 \dots

解答ステップ

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

以下の部分分数を得る:

\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

総和規則を適用する:

\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))

簡素化

- 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)