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受欢迎的微积分 >

integral from 0 to 0.6 of 1/((1-x)(3-x))

解答

\int_{0}^{0.6} \frac{1}{( 1 - x ) ( 3 - x )} d x

解答

0.34657 \dots

求解步骤

\int_{0}^{0.6} \frac{1}{( 1 - x ) ( 3 - x )} d x

对

(1 - x) (3 - x)

配方:

(x - 2)^{2} - 1

= \int_{0}^{0.6} \frac{1}{( x - 2 ) ^{2} - 1} d x

使用换元积分法

= \int_{- 2}^{- 1.4} \frac{1}{u ^{2} - 1} d u

因式分解

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

= \int_{- 2}^{- 1.4} \frac{1}{- ( - u ^{2} + 1 )} d u

提出常数:

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

= - \int_{- 2}^{- 1.4} \frac{1}{- u ^{2} + 1} d u

使用积分常见定则:

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

= - [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{- 2}^{- 1.4}

化简

- [\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2}]_{- 2}^{- 1.4} : - [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{- 2}^{- 1.4}

= - [\frac{1}{2} (ln ∣ u + 1 ∣ - ln ∣ u - 1 ∣)]_{- 2}^{- 1.4}

带入上下限计算后得:

- 0.34657 \dots

= - (- 0.34657 \dots)

化简

= 0.34657 \dots

作图

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