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

解答

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

2 (\frac{x ^{3}}{3 ( 1 + x ^{3} )} + \frac{1}{3} (ln x^{3} + 1 + \frac{1}{x ^{3} + 1})) + C

求解步骤

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

提出常数:

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

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

使用分布积分法

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

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

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

化简

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

解答补常数

= 2 (\frac{x ^{3}}{3 ( 1 + x ^{3} )} + \frac{1}{3} (ln x^{3} + 1 + \frac{1}{x ^{3} + 1})) + C

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