integral of (x^2)/(\sqrt[3]{x-1)}

解答

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

x^{2} (x - 1)^{\frac{2}{3}} - \frac{1}{40} (25 x^{2} - 18 x - 27) (x - 1)^{\frac{2}{3}} + C

求解步骤

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

使用换元积分法

= \int \frac{( u + 1 ) ^{2}}{3 u} d u

使用分布积分法

= u^{\frac{2}{3}} (u + 1)^{2} - \int \frac{6 3 u u ^{\frac{5}{3}} + 6 3 u u ^{\frac{2}{3}} - u ^{2} - 2 u - 1}{3 u ^{\frac{1}{3}}} d u

\int \frac{6 3 u u ^{\frac{5}{3}} + 6 3 u u ^{\frac{2}{3}} - u ^{2} - 2 u - 1}{3 u ^{\frac{1}{3}}} d u = \frac{1}{40} u^{\frac{2}{3}} (25 u^{2} + 32 u - 20)

= u^{\frac{2}{3}} (u + 1)^{2} - \frac{1}{40} u^{\frac{2}{3}} (25 u^{2} + 32 u - 20)

u = x - 1

代回

= (x - 1)^{\frac{2}{3}} (x - 1 + 1)^{2} - \frac{1}{40} (x - 1)^{\frac{2}{3}} (25 (x - 1)^{2} + 32 (x - 1) - 20)

化简

(x - 1)^{\frac{2}{3}} (x - 1 + 1)^{2} - \frac{1}{40} (x - 1)^{\frac{2}{3}} (25 (x - 1)^{2} + 32 (x - 1) - 20) : x^{2} (x - 1)^{\frac{2}{3}} - \frac{1}{40} (25 x^{2} - 18 x - 27) (x - 1)^{\frac{2}{3}}

= x^{2} (x - 1)^{\frac{2}{3}} - \frac{1}{40} (25 x^{2} - 18 x - 27) (x - 1)^{\frac{2}{3}}

解答补常数

= x^{2} (x - 1)^{\frac{2}{3}} - \frac{1}{40} (25 x^{2} - 18 x - 27) (x - 1)^{\frac{2}{3}} + C

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