integral of (x^2-1)/(x^2-4)

解

\int \frac{x ^{2} - 1}{x ^{2} - 4} d x

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

解答ステップ

\int \frac{x ^{2} - 1}{x ^{2} - 4} d x

\frac{x ^{2} - 1}{x ^{2} - 4} = \frac{3}{x ^{2} - 4} + 1

= \int \frac{3}{x ^{2} - 4} + 1 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

= \int \frac{3}{x ^{2} - 4} d x + \int 1 d x

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

\int 1 d x = x

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

定数を解答に追加する

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