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受欢迎的微积分 >

integral from 3 to 4 of 1/(x^2-3x+2)

解答

\int_{3}^{4} \frac{1}{x ^{2} - 3 x + 2} d x

解答

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

+1

十进制

0.28768 \dots

求解步骤

\int_{3}^{4} \frac{1}{x ^{2} - 3 x + 2} d x

对

x^{2} - 3 x + 2

配方:

(x - \frac{3}{2})^{2} - \frac{1}{4}

= \int_{3}^{4} \frac{1}{( x - \frac{3}{2} ) ^{2} - \frac{1}{4}} d x

使用换元积分法

= \int_{\frac{3}{2}}^{\frac{5}{2}} \frac{4}{4 u ^{2} - 1} d u

提出常数:

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

= 4 \cdot \int_{\frac{3}{2}}^{\frac{5}{2}} \frac{1}{4 u ^{2} - 1} d u

因式分解

4 u^{2} - 1 : - (- 4 u^{2} + 1)

= 4 \cdot \int_{\frac{3}{2}}^{\frac{5}{2}} \frac{1}{- ( - 4 u ^{2} + 1 )} d u

提出常数:

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

= 4 (- \int_{\frac{3}{2}}^{\frac{5}{2}} \frac{1}{- 4 u ^{2} + 1} d u)

使用换元积分法

= 4 (- \int_{3}^{5} \frac{1}{2 ( - v ^{2} + 1 )} d v)

提出常数:

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

= 4 (- \frac{1}{2} \cdot \int_{3}^{5} \frac{1}{- v ^{2} + 1} d v)

使用积分常见定则:

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

= 4 (- \frac{1}{2} [\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}]_{3}^{5})

化简

4 (- \frac{1}{2} [\frac{ln ∣ v + 1 ∣}{2} - \frac{ln ∣ v - 1 ∣}{2}]_{3}^{5}) : - 2 [\frac{1}{2} (ln ∣ v + 1 ∣ - ln ∣ v - 1 ∣)]_{3}^{5}

= - 2 [\frac{1}{2} (ln ∣ v + 1 ∣ - ln ∣ v - 1 ∣)]_{3}^{5}

带入上下限计算后得:

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

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

化简

= - ln (\frac{3}{4})

作图

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