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受欢迎的微积分 >

integral of (sqrt(x))/(sqrt(x)+1)

解答

\int \frac{x}{x + 1} d x

解答

x - 2 x + 2 ln x + 1 - 3 + C

求解步骤

\int \frac{x}{x + 1} d x

使用换元积分法

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

提出常数:

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

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

乘开

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

= 2 \cdot \int u - 2 + \frac{1}{u} 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 u d u - \int 2 d u + \int \frac{1}{u} d u)

\int u d u = \frac{u ^{2}}{2}

\int 2 d u = 2 u

\int \frac{1}{u} d u = ln ∣ u ∣

= 2 (\frac{u ^{2}}{2} - 2 u + ln ∣ u ∣)

u = x + 1

代回

= 2 (\frac{( x + 1 ) ^{2}}{2} - 2 (x + 1) + ln x + 1)

化简

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

= x - 2 x + 2 ln x + 1 - 3

解答补常数

= x - 2 x + 2 ln x + 1 - 3 + C

作图

流行的例子

integral of 1/((10+x)^2)

\int \frac{1}{( 10 + x ) ^{2}} d x

integral of x^3sqrt(3-x^2)

\int x^{3} 3 - x^{2} d x

derivative of (x^6/(1-x^5))

\frac{d}{d x} (\frac{x ^{6}}{1 - x ^{5}})

integral of 4/(sqrt(x^2+9))

\int \frac{4}{x ^{2} + 9} d x

integral from 0 to 5 of 1/4 (5-x)sqrt(x)

\int_{0}^{5} \frac{1}{4} (5 - x) x d x