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

解

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

\frac{3}{2} ln 4 x^{2} - 4 x - 12 + \frac{1}{2 13} ln \frac{1}{13} (2 x - 1) + 1 - \frac{1}{2 13} ln \frac{1}{13} (2 x - 1) - 1 + C

解答ステップ

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

拡張

\frac{3 x - 2}{x ^{2} - x - 3} : \frac{3 x}{x ^{2} - x - 3} - \frac{2}{x ^{2} - x - 3}

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

\int \frac{3 x}{x ^{2} - x - 3} d x = 6 (\frac{1}{4} ln 4 x^{2} - 4 x - 12 - \frac{1}{4 13} (ln \frac{2 x - 1}{13} + 1 - ln \frac{2 x - 1}{13} - 1))

\int \frac{2}{x ^{2} - x - 3} d x = - \frac{2}{13} (ln \frac{2 x - 1}{13} + 1 - ln \frac{2 x - 1}{13} - 1)

= 6 (\frac{1}{4} ln 4 x^{2} - 4 x - 12 - \frac{1}{4 13} (ln \frac{2 x - 1}{13} + 1 - ln \frac{2 x - 1}{13} - 1)) - (- \frac{2}{13} (ln \frac{2 x - 1}{13} + 1 - ln \frac{2 x - 1}{13} - 1))

簡素化

6 (\frac{1}{4} ln 4 x^{2} - 4 x - 12 - \frac{1}{4 13} (ln \frac{2 x - 1}{13} + 1 - ln \frac{2 x - 1}{13} - 1)) - (- \frac{2}{13} (ln \frac{2 x - 1}{13} + 1 - ln \frac{2 x - 1}{13} - 1)) : \frac{3}{2} ln 4 x^{2} - 4 x - 12 + \frac{1}{2 13} ln \frac{1}{13} (2 x - 1) + 1 - \frac{1}{2 13} ln \frac{1}{13} (2 x - 1) - 1

= \frac{3}{2} ln 4 x^{2} - 4 x - 12 + \frac{1}{2 13} ln \frac{1}{13} (2 x - 1) + 1 - \frac{1}{2 13} ln \frac{1}{13} (2 x - 1) - 1

定数を解答に追加する

= \frac{3}{2} ln 4 x^{2} - 4 x - 12 + \frac{1}{2 13} ln \frac{1}{13} (2 x - 1) + 1 - \frac{1}{2 13} ln \frac{1}{13} (2 x - 1) - 1 + C