integral of (ln(x+1))/(\sqrt[3]{x+1)}

解答

\int \frac{ln ( x + 1 )}{3 x + 1} d x

\frac{3}{2} (x + 1)^{\frac{2}{3}} ln (x + 1) - \frac{9}{4} (x + 1)^{\frac{2}{3}} + C

求解步骤

\int \frac{ln ( x + 1 )}{3 x + 1} d x

使用换元积分法

= \int \frac{ln ( u )}{3 u} d u

使用分布积分法

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

\int \frac{3}{2 u ^{\frac{1}{3}}} d u = \frac{9}{4} u^{\frac{2}{3}}

= \frac{3}{2} u^{\frac{2}{3}} ln (u) - \frac{9}{4} u^{\frac{2}{3}}

u = x + 1

代回

= \frac{3}{2} (x + 1)^{\frac{2}{3}} ln (x + 1) - \frac{9}{4} (x + 1)^{\frac{2}{3}}

解答补常数

= \frac{3}{2} (x + 1)^{\frac{2}{3}} ln (x + 1) - \frac{9}{4} (x + 1)^{\frac{2}{3}} + C

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