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人気のある微分積分 >

integral of 5/(2x^2-3x-2)

解

\int \frac{5}{2 x ^{2} - 3 x - 2} d x

解

- ln ∣ 2 x + 1 ∣ + ln ∣ x - 2 ∣ + C

解答ステップ

\int \frac{5}{2 x ^{2} - 3 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 \frac{1}{2 x ^{2} - 3 x - 2} d x

以下の部分分数を得る:

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

= 5 \cdot \int - \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 \frac{2}{5 ( 2 x + 1 )} d x + \int \frac{1}{5 ( x - 2 )} d x)

\int \frac{2}{5 ( 2 x + 1 )} d x = \frac{1}{5} ln ∣ 2 x + 1 ∣

\int \frac{1}{5 ( x - 2 )} d x = \frac{1}{5} ln ∣ x - 2 ∣

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

簡素化

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

= - ln ∣ 2 x + 1 ∣ + ln ∣ x - 2 ∣

定数を解答に追加する

= - ln ∣ 2 x + 1 ∣ + ln ∣ x - 2 ∣ + C

グラフ

人気の例

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\int - ln (x) x d x

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