integral from 1 to 4 of (18(ln(x))^5)/x

解答

\int_{1}^{4} \frac{18 ( ln ( x ) ) ^{5}}{x} d x

192 ln^{6} (2)

十进制

21.29384 \dots

求解步骤

\int_{1}^{4} \frac{18 ( ln ( x ) ) ^{5}}{x} d x

提出常数:

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

= 18 \cdot \int_{1}^{4} \frac{( ln ( x ) ) ^{5}}{x} d x

使用换元积分法

= 18 \cdot \int_{0}^{2 l n (2)} u^{5} d u

使用幂法则

= 18 [\frac{u ^{6}}{6}]_{0}^{2 l n (2)}

带入上下限计算后得:

\frac{32 ln ^{6} ( 2 )}{3}

= 18 \cdot \frac{32 ln ^{6} ( 2 )}{3}

化简

= 192 ln^{6} (2)

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