integral from 1 to 4 of (x-2)/(x(x+1))

解

\int_{1}^{4} \frac{x - 2}{x ( x + 1 )} d x

- 7 ln (2) + 3 ln (5)

十進法表記

- 0.02371 \dots

解答ステップ

\int_{1}^{4} \frac{x - 2}{x ( x + 1 )} d x

拡張

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

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_{1}^{4} \frac{x}{x ( x + 1 )} d x - \int_{1}^{4} \frac{2}{x ( x + 1 )} d x

\int_{1}^{4} \frac{x}{x ( x + 1 )} d x = ln (5) - ln (2)

\int_{1}^{4} \frac{2}{x ( x + 1 )} d x = 2 (3 ln (2) - ln (5))

= ln (5) - ln (2) - 2 (3 ln (2) - ln (5))

簡素化

ln (5) - ln (2) - 2 (3 ln (2) - ln (5)) : - 7 ln (2) + 3 ln (5)

= - 7 ln (2) + 3 ln (5)