integral of (x+1)/(x^2(x^2+1))

解答

\int \frac{x + 1}{x ^{2} ( x ^{2} + 1 )} d x

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

求解步骤

\int \frac{x + 1}{x ^{2} ( x ^{2} + 1 )} d x

使用分布积分法

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

化简

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

解答补常数

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

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