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受欢迎的微积分 >

integral from 9 to 16 of 1/(6-sqrt(x))

解答

\int_{9}^{16} \frac{1}{6 - x} d x

解答

2 (- 1 + 6 (ln (3) - ln (2)))

+1

十进制

2.86558 \dots

求解步骤

\int_{9}^{16} \frac{1}{6 - x} d x

使用换元积分法

= \int_{3}^{4} \frac{2 u}{6 - u} d u

提出常数:

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

= 2 \cdot \int_{3}^{4} \frac{u}{6 - u} d u

使用换元积分法

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

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

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

提出常数:

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

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

乘开

\frac{- v + 6}{v} : - 1 + \frac{6}{v}

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

\int_{2}^{3} 1 d v = 1

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

= 2 (- (- (- 1 + 6 (ln (3) - ln (2)))))

化简

= 2 (- 1 + 6 (ln (3) - ln (2)))

作图

流行的例子

derivative of (ln(x-10)/(ln(10)))

\frac{d}{d x} (\frac{ln ( x ) - 10}{ln ( 10 )})

integral from 2 to e^4 of 1/x

\int_{2}^{e^{4}} \frac{1}{x} d x

inverselaplace 4/(s^4)

in v erse l a pl a ce \frac{4}{s ^{4}}

integral of x^7e^{3x^4}

\int x^{7} e^{3 x^{4}} d x

derivative (t-sqrt(t))/(t^{1/9)}

d er i v a t i v e \frac{t - t}{t ^{\frac{1}{9}}}