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受欢迎的微积分 >

integral from 1 to 16 of 1/(7-sqrt(x))

解答

\int_{1}^{16} \frac{1}{7 - x} d x

解答

2 (- 3 + 7 ln (2))

+1

十进制

3.70406 \dots

求解步骤

\int_{1}^{16} \frac{1}{7 - x} d x

使用换元积分法

= \int_{1}^{4} \frac{2 u}{7 - u} d u

提出常数:

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

= 2 \cdot \int_{1}^{4} \frac{u}{7 - u} d u

使用换元积分法

= 2 \cdot \int_{6}^{3} - \frac{- v + 7}{v} d v

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 2 (- \int_{3}^{6} - \frac{- v + 7}{v} d v)

提出常数:

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

= 2 (- (- \int_{3}^{6} \frac{- v + 7}{v} d v))

乘开

\frac{- v + 7}{v} : - 1 + \frac{7}{v}

= 2 (- (- \int_{3}^{6} - 1 + \frac{7}{v} d v))

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

= 2 (- (- (- \int_{3}^{6} 1 d v + \int_{3}^{6} \frac{7}{v} d v)))

\int_{3}^{6} 1 d v = 3

\int_{3}^{6} \frac{7}{v} d v = 7 ln (2)

= 2 (- (- (- 3 + 7 ln (2))))

化简

= 2 (- 3 + 7 ln (2))

作图

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