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受欢迎的微积分 >

integral from 1 to e^2 of (ln^8(x^2))/x

解答

\int_{1}^{e^{2}} \frac{ln ^{8} ( x ^{2} )}{x} d x

解答

\frac{131072}{9}

+1

十进制

14563.55555 \dots

求解步骤

\int_{1}^{e^{2}} \frac{ln ^{8} ( x ^{2} )}{x} d x

化简

\frac{ln ^{8} ( x ^{2} )}{x} : \frac{256 ln ^{8} ( x )}{x}

= \int_{1}^{e^{2}} \frac{256 ln ^{8} ( x )}{x} d x

提出常数:

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

= 256 \cdot \int_{1}^{e^{2}} \frac{ln ^{8} ( x )}{x} d x

使用换元积分法

= 256 \cdot \int_{0}^{2} u^{8} d u

使用幂法则

= 256 [\frac{u ^{9}}{9}]_{0}^{2}

带入上下限计算后得:

\frac{512}{9}

= 256 \cdot \frac{512}{9}

化简

= \frac{131072}{9}

作图

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(\partial)/(\partial x)(e^{kx-mt})

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