integral from 3 to 6 of x/(x^2+x-2)

解

\int_{3}^{6} \frac{x}{x ^{2} + x - 2} d x

2 ln (2) + \frac{- ln ( 5 ) - ln ( 2 )}{3}

十進法表記

0.61876 \dots

解答ステップ

\int_{3}^{6} \frac{x}{x ^{2} + x - 2} d x

以下の部分分数を得る:

\frac{x}{x ^{2} + x - 2} : \frac{1}{3 ( x - 1 )} + \frac{2}{3 ( x + 2 )}

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

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

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

= \frac{1}{3} (ln (5) - ln (2)) + 2 ln (2) - \frac{2}{3} ln (5)

簡素化

\frac{1}{3} (ln (5) - ln (2)) + 2 ln (2) - \frac{2}{3} ln (5) : 2 ln (2) + \frac{- ln ( 5 ) - ln ( 2 )}{3}

= 2 ln (2) + \frac{- ln ( 5 ) - ln ( 2 )}{3}