integral of (x^2)/(x^4-42x^2-343)

解

\int \frac{x ^{2}}{x ^{4} - 42 x ^{2} - 343} d x

\frac{1}{8 7} arctan (\frac{x}{7}) - \frac{1}{16} ln ∣ x + 7 ∣ + \frac{1}{16} ln ∣ x - 7 ∣ + C

解答ステップ

\int \frac{x ^{2}}{x ^{4} - 42 x ^{2} - 343} d x

以下の部分分数を得る:

\frac{x ^{2}}{x ^{4} - 42 x ^{2} - 343} : \frac{1}{8 ( x ^{2} + 7 )} - \frac{1}{16 ( x + 7 )} + \frac{1}{16 ( x - 7 )}

= \int \frac{1}{8 ( x ^{2} + 7 )} - \frac{1}{16 ( x + 7 )} + \frac{1}{16 ( x - 7 )} 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{1}{8 ( x ^{2} + 7 )} d x - \int \frac{1}{16 ( x + 7 )} d x + \int \frac{1}{16 ( x - 7 )} d x

\int \frac{1}{8 ( x ^{2} + 7 )} d x = \frac{1}{8 7} arctan (\frac{x}{7})

\int \frac{1}{16 ( x + 7 )} d x = \frac{1}{16} ln ∣ x + 7 ∣

\int \frac{1}{16 ( x - 7 )} d x = \frac{1}{16} ln ∣ x - 7 ∣

= \frac{1}{8 7} arctan (\frac{x}{7}) - \frac{1}{16} ln ∣ x + 7 ∣ + \frac{1}{16} ln ∣ x - 7 ∣

定数を解答に追加する

= \frac{1}{8 7} arctan (\frac{x}{7}) - \frac{1}{16} ln ∣ x + 7 ∣ + \frac{1}{16} ln ∣ x - 7 ∣ + C