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受欢迎的微积分 >

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

解答

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

解答

\frac{1}{6} ln^{2} (1 + x^{3}) + C

求解步骤

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

使用换元积分法

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

\frac{ln ( 1 + u )}{1 + u} = ln (1 + u) \frac{1}{1 + u}

= \frac{1}{3} \cdot \int ln (1 + u) \frac{1}{1 + u} d u

使用换元积分法

= \frac{1}{3} \cdot \int v d v

使用幂法则

= \frac{1}{3} \cdot \frac{v ^{2}}{2}

代回

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

化简

\frac{1}{3} \cdot \frac{ln ^{2} ( 1 + x ^{3} )}{2} : \frac{1}{6} ln^{2} (1 + x^{3})

= \frac{1}{6} ln^{2} (1 + x^{3})

解答补常数

= \frac{1}{6} ln^{2} (1 + x^{3}) + C

作图

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