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受欢迎的微积分 >

integral from 3 to 8 of (sqrt(x+1))/x

解答

\int_{3}^{8} \frac{x + 1}{x} d x

解答

- ln (2) + ln (3) + 2

+1

十进制

2.40546 \dots

求解步骤

\int_{3}^{8} \frac{x + 1}{x} d x

使用换元积分法

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

提出常数:

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

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

因式分解

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

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

提出常数:

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

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

\frac{u ^{2}}{- u ^{2} + 1} = \frac{1}{- u ^{2} + 1} - 1

= 2 (- \int_{2}^{3} \frac{1}{- u ^{2} + 1} - 1 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

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

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

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

= 2 (- (\frac{ln ( 2 ) - ln ( 3 )}{2} - 1))

化简

2 (- (\frac{ln ( 2 ) - ln ( 3 )}{2} - 1)) : - ln (2) + ln (3) + 2

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

作图

流行的例子

面积 y=3x^2,y=6-3x

a re a y = 3 x^{2}, y = 6 - 3 x

面积 y=x,y=5x,y=-x+4

a re a y = x, y = 5 x, y = - x + 4

integral of (2x+1)/(x^2+2x+17)

\int \frac{2 x + 1}{x ^{2} + 2 x + 17} d x

integral from 0 to 1 of 13cos(x^2)

\int_{0}^{1} 13 cos (x^{2}) d x

derivative-csc(x)

d er i v a t i v e - csc (x)