integral of (6x^3-12x^2+7)/(x^2-2x)

解

\int \frac{6 x ^{3} - 12 x ^{2} + 7}{x ^{2} - 2 x} d x

3 x^{2} + \frac{7}{2} ln ∣ x - 2 ∣ - \frac{7}{2} ln ∣ x ∣ - 12 + C

解答ステップ

\int \frac{6 x ^{3} - 12 x ^{2} + 7}{x ^{2} - 2 x} d x

拡張

\frac{6 x ^{3} - 12 x ^{2} + 7}{x ^{2} - 2 x} : \frac{6 x ^{3}}{x ^{2} - 2 x} - \frac{12 x ^{2}}{x ^{2} - 2 x} + \frac{7}{x ^{2} - 2 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

= \int \frac{6 x ^{3}}{x ^{2} - 2 x} d x - \int \frac{12 x ^{2}}{x ^{2} - 2 x} d x + \int \frac{7}{x ^{2} - 2 x} d x

\int \frac{6 x ^{3}}{x ^{2} - 2 x} d x = 6 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣)

\int \frac{12 x ^{2}}{x ^{2} - 2 x} d x = 12 (x - 2 + 2 ln ∣ x - 2 ∣)

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

= 6 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) - 12 (x - 2 + 2 ln ∣ x - 2 ∣) - 7 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣)

簡素化

6 (\frac{( x - 2 ) ^{2}}{2} + 4 (x - 2) + 4 ln ∣ x - 2 ∣) - 12 (x - 2 + 2 ln ∣ x - 2 ∣) - 7 (\frac{1}{2} ln ∣ x ∣ - \frac{1}{2} ln ∣ x - 2 ∣) : 3 x^{2} + \frac{7}{2} ln ∣ x - 2 ∣ - \frac{7}{2} ln ∣ x ∣ - 12

= 3 x^{2} + \frac{7}{2} ln ∣ x - 2 ∣ - \frac{7}{2} ln ∣ x ∣ - 12

定数を解答に追加する

= 3 x^{2} + \frac{7}{2} ln ∣ x - 2 ∣ - \frac{7}{2} ln ∣ x ∣ - 12 + C