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受欢迎的微积分 >

integral from 1 to 2 of (4ln(x))/(x^2)

解答

\int_{1}^{2} \frac{4 ln ( x )}{x ^{2}} d x

解答

- 2 ln (2) + 2

+1

十进制

0.61370 \dots

求解步骤

\int_{1}^{2} \frac{4 ln ( x )}{x ^{2}} d x

提出常数:

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

= 4 \cdot \int_{1}^{2} \frac{ln ( x )}{x ^{2}} d x

使用分布积分法

= 4 [- \frac{ln ( x )}{x} - \int - \frac{1}{x ^{2}} d x]_{1}^{2}

\int - \frac{1}{x ^{2}} d x = \frac{1}{x}

= 4 [- \frac{ln ( x )}{x} - \frac{1}{x}]_{1}^{2}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= 4 [\frac{- ln ( x ) - 1}{x}]_{1}^{2}

带入上下限计算后得:

\frac{- ln ( 2 ) - 1}{2} + 1

= 4 (\frac{- ln ( 2 ) - 1}{2} + 1)

化简

= - 2 ln (2) + 2

作图

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