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人気のある微分積分 >

integral from 3 to 5 of 5/((2x+1)(x-2))

解

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

解

- ln (11) + ln (7) + ln (3)

+1

十進法表記

0.64662 \dots

解答ステップ

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

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

以下の部分分数を得る:

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

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

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

\int_{3}^{5} \frac{2}{5 ( 2 x + 1 )} d x = \frac{1}{5} (ln (11) - ln (7))

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

= 5 (- \frac{1}{5} (ln (11) - ln (7)) + \frac{1}{5} ln (3))

簡素化

5 (- \frac{1}{5} (ln (11) - ln (7)) + \frac{1}{5} ln (3)) : - ln (11) + ln (7) + ln (3)

= - ln (11) + ln (7) + ln (3)

グラフ

人気の例

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