integral from 3 to 7 of-2log_{10}(x)

解答

\int_{3}^{7} - 2 lo g_{10} (x) d x

- \frac{2 ( 7 ln ( 7 ) - 4 - 3 ln ( 3 ) )}{ln ( 10 )}

十进制

- 5.49428 \dots

求解步骤

\int_{3}^{7} - 2 lo g_{10} (x) d x

提出常数:

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

= - 2 \cdot \int_{3}^{7} lo g_{10} (x) d x

使用分布积分法

= - 2 [x lo g_{10} (x) - \int \frac{1}{ln ( 10 )} d x]_{3}^{7}

\int \frac{1}{ln ( 10 )} d x = \frac{1}{ln ( 10 )} x

= - 2 [x lo g_{10} (x) - \frac{1}{ln ( 10 )} x]_{3}^{7}

化简

x lo g_{10} (x) - \frac{1}{ln ( 10 )} x : \frac{- x + x ln ( x )}{ln ( 10 )}

= - 2 [\frac{- x + x ln ( x )}{ln ( 10 )}]_{3}^{7}

带入上下限计算后得:

\frac{7 ln ( 7 ) - 4 - 3 ln ( 3 )}{ln ( 10 )}

= - 2 \cdot \frac{7 ln ( 7 ) - 4 - 3 ln ( 3 )}{ln ( 10 )}

化简

= - \frac{2 ( 7 ln ( 7 ) - 4 - 3 ln ( 3 ) )}{ln ( 10 )}

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