integral of (3x^2-3x+1)/(x^3+x)

解答

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

ln x^{2} + 1 - 3 arctan (x) + ln ∣ x ∣ + C

求解步骤

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

乘开

\frac{3 x ^{2} - 3 x + 1}{x ^{3} + x} : \frac{3 x ^{2}}{x ^{3} + x} - \frac{3 x}{x ^{3} + x} + \frac{1}{x ^{3} + x}

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

\int \frac{3 x}{x ^{3} + x} d x = 3 arctan (x)

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

= \frac{3}{2} ln x^{2} + 1 - 3 arctan (x) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1

化简

\frac{3}{2} ln x^{2} + 1 - 3 arctan (x) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 1 : ln x^{2} + 1 - 3 arctan (x) + ln ∣ x ∣

= ln x^{2} + 1 - 3 arctan (x) + ln ∣ x ∣

解答补常数

= ln x^{2} + 1 - 3 arctan (x) + ln ∣ x ∣ + C

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