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人気のある微分積分 >

integral of 7/(x^2+3x-10)

解

\int \frac{7}{x ^{2} + 3 x - 10} d x

解

ln ∣ x - 2 ∣ - ln ∣ x + 5 ∣ + C

解答ステップ

\int \frac{7}{x ^{2} + 3 x - 10} d x

定数を除く:

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

= 7 \cdot \int \frac{1}{x ^{2} + 3 x - 10} d x

以下の部分分数を得る:

\frac{1}{x ^{2} + 3 x - 10} : \frac{1}{7 ( x - 2 )} - \frac{1}{7 ( x + 5 )}

= 7 \cdot \int \frac{1}{7 ( x - 2 )} - \frac{1}{7 ( 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

= 7 (\int \frac{1}{7 ( x - 2 )} d x - \int \frac{1}{7 ( x + 5 )} d x)

\int \frac{1}{7 ( x - 2 )} d x = \frac{1}{7} ln ∣ x - 2 ∣

\int \frac{1}{7 ( x + 5 )} d x = \frac{1}{7} ln ∣ x + 5 ∣

= 7 (\frac{1}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln ∣ x + 5 ∣)

簡素化

7 (\frac{1}{7} ln ∣ x - 2 ∣ - \frac{1}{7} ln ∣ x + 5 ∣) : ln ∣ x - 2 ∣ - ln ∣ x + 5 ∣

= ln ∣ x - 2 ∣ - ln ∣ x + 5 ∣

定数を解答に追加する

= ln ∣ x - 2 ∣ - ln ∣ x + 5 ∣ + C

グラフ

人気の例

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