integral from 6 to 8 of 1/(x^2-1)

解答

\int_{6}^{8} \frac{1}{x ^{2} - 1} d x

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

十进制

0.04257 \dots

求解步骤

\int_{6}^{8} \frac{1}{x ^{2} - 1} d x

因式分解

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

= \int_{6}^{8} \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

= - \int_{6}^{8} \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}

= - [\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2}]_{6}^{8}

化简

- [\frac{ln ∣ x + 1 ∣}{2} - \frac{ln ∣ x - 1 ∣}{2}]_{6}^{8} : - [\frac{1}{2} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣)]_{6}^{8}

= - [\frac{1}{2} (ln ∣ x + 1 ∣ - ln ∣ x - 1 ∣)]_{6}^{8}

带入上下限计算后得:

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

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

x \to 0 lim (x sec (x) csc (x))

t an g e n t y = \frac{7}{x}, (4, \frac{7}{2})

\frac{d}{d x} (50 - x^{2} - x)

\frac{d}{d x} (64)

\int \frac{- 1000}{x ^{2}} d x