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人気のある微分積分 >

integral of (x^2+9x+2)/((x-1)(x+1)(x+2))

解

\int \frac{x ^{2} + 9 x + 2}{( x - 1 ) ( x + 1 ) ( x + 2 )} d x

解

\frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣ + \frac{7}{3} ln ∣ x - 1 ∣ - \frac{7}{3} ln ∣ x + 2 ∣ + 3 ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣ + C

解答ステップ

\int \frac{x ^{2} + 9 x + 2}{( x - 1 ) ( x + 1 ) ( x + 2 )} d x

u置換積分法を適用する

= \int \frac{u ^{2} + 7 u - 6}{u ( u - 2 ) ( u + 1 )} d u

拡張

\frac{u ^{2} + 7 u - 6}{u ( u - 2 ) ( u + 1 )} : \frac{u ^{2}}{u ( u - 2 ) ( u + 1 )} + \frac{7 u}{u ( u - 2 ) ( u + 1 )} - \frac{6}{u ( u - 2 ) ( u + 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{u ^{2}}{u ( u - 2 ) ( u + 1 )} d u + \int \frac{7 u}{u ( u - 2 ) ( u + 1 )} d u - \int \frac{6}{u ( u - 2 ) ( u + 1 )} d u

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

\int \frac{7 u}{u ( u - 2 ) ( u + 1 )} d u = 7 (\frac{1}{3} ln ∣ u - 2 ∣ - \frac{1}{3} ln ∣ u + 1 ∣)

\int \frac{6}{u ( u - 2 ) ( u + 1 )} d u = - 3 ln ∣ u ∣ + ln ∣ u - 2 ∣ + 2 ln ∣ u + 1 ∣

= \frac{2}{3} ln ∣ u - 2 ∣ + \frac{1}{3} ln ∣ u + 1 ∣ + 7 (\frac{1}{3} ln ∣ u - 2 ∣ - \frac{1}{3} ln ∣ u + 1 ∣) - (- 3 ln ∣ u ∣ + ln ∣ u - 2 ∣ + 2 ln ∣ u + 1 ∣)

代用を戻す

u = x + 1

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

簡素化

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

= \frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣ + \frac{7}{3} ln ∣ x - 1 ∣ - \frac{7}{3} ln ∣ x + 2 ∣ + 3 ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣

定数を解答に追加する

= \frac{2}{3} ln ∣ x - 1 ∣ + \frac{1}{3} ln ∣ x + 2 ∣ + \frac{7}{3} ln ∣ x - 1 ∣ - \frac{7}{3} ln ∣ x + 2 ∣ + 3 ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣ - 2 ln ∣ x + 2 ∣ + C

グラフ

人気の例

integral of sin(0.01t)+t^3

\int sin (0.01 t) + t^{3} d t

integral of x(7-x^2)

\int x (7 - x^{2}) d x

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\int lo g_{3} (6 x^{2}) x d x

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\int \frac{4}{x ^{2} + 4} d x