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受欢迎的微积分 >

integral of 1/((60-x)(150-x))

解答

\int \frac{1}{( 60 - x ) ( 150 - x )} d x

解答

- \frac{1}{90} ln ∣ 60 - x ∣ + \frac{1}{90} ln ∣ - x + 150 ∣ + C

求解步骤

\int \frac{1}{( 60 - x ) ( 150 - x )} d x

使用换元积分法

= \int - \frac{1}{u ( u + 90 )} d u

提出常数:

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

= - \int \frac{1}{u ( u + 90 )} d u

将

\frac{1}{u ( u + 90 )}

用部份分式展开:

\frac{1}{90 u} - \frac{1}{90 ( u + 90 )}

= - \int \frac{1}{90 u} - \frac{1}{90 ( u + 90 )} d u

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 \frac{1}{90 u} d u - \int \frac{1}{90 ( u + 90 )} d u)

\int \frac{1}{90 u} d u = \frac{1}{90} ln ∣ u ∣

\int \frac{1}{90 ( u + 90 )} d u = \frac{1}{90} ln ∣ u + 90 ∣

= - (\frac{1}{90} ln ∣ u ∣ - \frac{1}{90} ln ∣ u + 90 ∣)

u = 60 - x

代回

= - (\frac{1}{90} ln ∣ 60 - x ∣ - \frac{1}{90} ln ∣ 60 - x + 90 ∣)

化简

- (\frac{1}{90} ln ∣ 60 - x ∣ - \frac{1}{90} ln ∣ 60 - x + 90 ∣) : - \frac{1}{90} ln ∣ 60 - x ∣ + \frac{1}{90} ln ∣ - x + 150 ∣

= - \frac{1}{90} ln ∣ 60 - x ∣ + \frac{1}{90} ln ∣ - x + 150 ∣

解答补常数

= - \frac{1}{90} ln ∣ 60 - x ∣ + \frac{1}{90} ln ∣ - x + 150 ∣ + C

作图

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