integral of (3x^2-5x+7)/(x+1)

解

\int \frac{3 x ^{2} - 5 x + 7}{x + 1} d x

- 11 x + 15 ln ∣ x + 1 ∣ + \frac{3}{2} (x + 1)^{2} - 11 + C

解答ステップ

\int \frac{3 x ^{2} - 5 x + 7}{x + 1} d x

拡張

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

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 \frac{3 x ^{2}}{x + 1} d x - \int \frac{5 x}{x + 1} d x + \int \frac{7}{x + 1} d x

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

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

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

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

簡素化

3 (\frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + ln ∣ x + 1 ∣) - 5 (x + 1 - ln ∣ x + 1 ∣) + 7 ln ∣ x + 1 ∣ : - 11 x + 15 ln ∣ x + 1 ∣ + \frac{3}{2} (x + 1)^{2} - 11

= - 11 x + 15 ln ∣ x + 1 ∣ + \frac{3}{2} (x + 1)^{2} - 11

定数を解答に追加する

= - 11 x + 15 ln ∣ x + 1 ∣ + \frac{3}{2} (x + 1)^{2} - 11 + C