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受欢迎的微积分 >

integral from 2 to 5 of 5/(x^2-1)

解答

\int_{2}^{5} \frac{5}{x ^{2} - 1} d x

解答

\frac{5 ln ( 2 )}{2}

+1

十进制

1.73286 \dots

求解步骤

\int_{2}^{5} \frac{5}{x ^{2} - 1} d x

提出常数:

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

= 5 \cdot \int_{2}^{5} \frac{1}{x ^{2} - 1} d x

因式分解

x^{2} - 1 : - (- x^{2} + 1)

= 5 \cdot \int_{2}^{5} \frac{1}{- ( - x ^{2} + 1 )} d x

提出常数:

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

= 5 (- \int_{2}^{5} \frac{1}{- x ^{2} + 1} d x)

使用积分常见定则:

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

= 5 (- [\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2}]_{2}^{5})

化简

5 (- [\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2}]_{2}^{5}) : - 5 [\frac{1}{2} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣)]_{2}^{5}

= - 5 [\frac{1}{2} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣)]_{2}^{5}

带入上下限计算后得:

\frac{ln ( 6 ) - 2 ln ( 2 ) - ln ( 3 )}{2}

= - 5 \cdot \frac{ln ( 6 ) - 2 ln ( 2 ) - ln ( 3 )}{2}

化简

= \frac{5 ln ( 2 )}{2}

作图

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