integral of (2x^5-3x^2)/(1+3x^3-x^6)

解答

\int \frac{2 x ^{5} - 3 x ^{2}}{1 + 3 x ^{3} - x ^{6}} d x

- \frac{1}{3} ln 1 + 3 x^{3} - x^{6} + C

求解步骤

\int \frac{2 x ^{5} - 3 x ^{2}}{1 + 3 x ^{3} - x ^{6}} d x

使用换元积分法

= \int - \frac{1}{3 u} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \frac{1}{3} \cdot \int \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{3} ln ∣ u ∣

u = 1 + 3 x^{3} - x^{6}

代回

= - \frac{1}{3} ln 1 + 3 x^{3} - x^{6}

解答补常数

= - \frac{1}{3} ln 1 + 3 x^{3} - x^{6} + C

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