integral from 5 to 10 of (5x-2)/(x^2-4)

解答

\int_{5}^{10} \frac{5 x - 2}{x ^{2} - 4} d x

\frac{5 ln ( \frac{32}{7} ) + 2 ln ( \frac{3}{2} ) - ln ( \frac{7}{2} )}{2}

十进制

3.57864 \dots

求解步骤

\int_{5}^{10} \frac{5 x - 2}{x ^{2} - 4} d x

乘开

\frac{5 x - 2}{x ^{2} - 4} : \frac{5 x}{x ^{2} - 4} - \frac{2}{x ^{2} - 4}

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{5}^{10} \frac{5 x}{x ^{2} - 4} d x - \int_{5}^{10} \frac{2}{x ^{2} - 4} d x

\int_{5}^{10} \frac{5 x}{x ^{2} - 4} d x = \frac{5}{2} ln (\frac{32}{7})

\int_{5}^{10} \frac{2}{x ^{2} - 4} d x = - \frac{2 ln ( \frac{3}{2} ) - ln ( \frac{7}{2} )}{2}

= \frac{5}{2} ln (\frac{32}{7}) - (- \frac{2 ln ( \frac{3}{2} ) - ln ( \frac{7}{2} )}{2})

化简

\frac{5}{2} ln (\frac{32}{7}) - (- \frac{2 ln ( \frac{3}{2} ) - ln ( \frac{7}{2} )}{2}) : \frac{5 ln ( \frac{32}{7} ) + 2 ln ( \frac{3}{2} ) - ln ( \frac{7}{2} )}{2}

= \frac{5 ln ( \frac{32}{7} ) + 2 ln ( \frac{3}{2} ) - ln ( \frac{7}{2} )}{2}

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