integral of (x^4)/(x^4-16)

解

\int \frac{x ^{4}}{x ^{4} - 16} d x

- arctan (\frac{x}{2}) - \frac{1}{2} ln ∣ x + 2 ∣ + \frac{1}{2} ln ∣ x - 2 ∣ + x + C

解答ステップ

\int \frac{x ^{4}}{x ^{4} - 16} d x

\frac{x ^{4}}{x ^{4} - 16} = \frac{16}{x ^{4} - 16} + 1

= \int \frac{16}{x ^{4} - 16} + 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{16}{x ^{4} - 16} d x + \int 1 d x

\int \frac{16}{x ^{4} - 16} d x = - arctan (\frac{x}{2}) - \frac{1}{2} ln ∣ x + 2 ∣ + \frac{1}{2} ln ∣ x - 2 ∣

\int 1 d x = x

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

定数を解答に追加する

= - arctan (\frac{x}{2}) - \frac{1}{2} ln ∣ x + 2 ∣ + \frac{1}{2} ln ∣ x - 2 ∣ + x + C