integral of (x^3ln(sqrt(x)))/3

解答

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

\frac{1}{3} (\frac{1}{8} x^{4} ln (x) - \frac{x ^{4}}{32}) + C

求解步骤

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

提出常数:

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

= \frac{1}{3} \cdot \int x^{3} ln (x) d x

使用分布积分法

= \frac{1}{3} (\frac{1}{4} x^{4} ln (x) - \int \frac{x ^{3}}{8} d x)

\int \frac{x ^{3}}{8} d x = \frac{x ^{4}}{32}

= \frac{1}{3} (\frac{1}{4} x^{4} ln (x) - \frac{x ^{4}}{32})

\frac{1}{4} x^{4} ln (x) = \frac{1}{8} x^{4} ln (x)

= \frac{1}{3} (\frac{1}{8} x^{4} ln (x) - \frac{x ^{4}}{32})

解答补常数

= \frac{1}{3} (\frac{1}{8} x^{4} ln (x) - \frac{x ^{4}}{32}) + C

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