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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

\int \frac{x}{1 - x} 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 = 1 - x

代回

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

化简

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

\int sec^{2} (x) - csc^{2} (x) d x

integral of cos^2(-x)

\int cos^{2} (- x) d x

integral of 1/(64sin^2(x)cos(x))

\int \frac{1}{64 sin ^{2} ( x ) cos ( x )} d x

integral of (sin(x))/(sqrt(cos(2x)))

\int \frac{sin ( x )}{cos ( 2 x )} d x

integral of (x+2)/(x^3+9x)

\int \frac{x + 2}{x ^{3} + 9 x} d x