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受欢迎的微积分 >

integral from 1 to e of ((ln(x^2))^2)/x

解答

\int_{1}^{e} \frac{( ln ( x ^{2} ) ) ^{2}}{x} d x

解答

\frac{4}{3}

+1

十进制

1.33333 \dots

求解步骤

\int_{1}^{e} \frac{( ln ( x ^{2} ) ) ^{2}}{x} d x

化简

= \int_{1}^{e} \frac{ln ^{2} ( x ^{2} )}{x} d x

使用换元积分法

= \int_{1}^{e^{2}} \frac{ln ^{2} ( u )}{2 u} d u

提出常数:

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

= \frac{1}{2} \cdot \int_{1}^{e^{2}} \frac{ln ^{2} ( u )}{u} d u

使用换元积分法

= \frac{1}{2} \cdot \int_{0}^{2} v^{2} d v

使用幂法则

= \frac{1}{2} [\frac{v ^{3}}{3}]_{0}^{2}

带入上下限计算后得:

\frac{8}{3}

= \frac{1}{2} \cdot \frac{8}{3}

化简

= \frac{4}{3}

作图

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