integral from 1 to 2 of 1/(x^2-4x-5)

解

\int_{1}^{2} \frac{1}{x ^{2} - 4 x - 5} d x

- \frac{1}{6} ln (2)

十進法表記

- 0.11552 \dots

解答ステップ

\int_{1}^{2} \frac{1}{x ^{2} - 4 x - 5} d x

以下の部分分数を得る:

\frac{1}{x ^{2} - 4 x - 5} : - \frac{1}{6 ( x + 1 )} + \frac{1}{6 ( x - 5 )}

= \int_{1}^{2} - \frac{1}{6 ( x + 1 )} + \frac{1}{6 ( x - 5 )} 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_{1}^{2} \frac{1}{6 ( x + 1 )} d x + \int_{1}^{2} \frac{1}{6 ( x - 5 )} d x

\int_{1}^{2} \frac{1}{6 ( x + 1 )} d x = \frac{1}{6} (ln (3) - ln (2))

\int_{1}^{2} \frac{1}{6 ( x - 5 )} d x = \frac{1}{6} (ln (3) - 2 ln (2))

= - \frac{1}{6} (ln (3) - ln (2)) + \frac{1}{6} (ln (3) - 2 ln (2))

簡素化

- \frac{1}{6} (ln (3) - ln (2)) + \frac{1}{6} (ln (3) - 2 ln (2)) : - \frac{1}{6} ln (2)

= - \frac{1}{6} ln (2)