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

解

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

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

解答ステップ

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

部分積分を適用する

= - 1 - x arctan (x) + \frac{2}{x} + 2 arctan (x) - \int - \frac{1}{x} - arctan (x) d x

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

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

簡素化

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

定数を解答に追加する

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