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受欢迎的微积分 >

integral of x/(1-x)

解答

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

解答

1 - x - ln ∣ 1 - x ∣ + C

求解步骤

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

使用换元积分法

= \int - \frac{- u + 1}{u} d u

提出常数:

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

= - \int \frac{- u + 1}{u} d u

乘开

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

= - \int - 1 + \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

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

\int 1 d u = u

\int \frac{1}{u} d u = ln ∣ u ∣

= - (- u + ln ∣ u ∣)

u = 1 - x

代回

= - (- (1 - x) + ln ∣ 1 - x ∣)

化简

= 1 - x - ln ∣ 1 - x ∣

解答补常数

= 1 - x - ln ∣ 1 - x ∣ + C

作图

流行的例子

derivative of 5(2xcos(7x-7x^2sin(7x)))

\frac{d}{d x} (5 (2 x cos (7 x) - 7 x^{2} sin (7 x)))

integral of cos^3(20x)

\int cos^{3} (20 x) d x

y^{''}+2y=sin(sqrt(2)x),y(0)=0,y^'(0)=0

y^{^{''}} + 2 y = sin (2 x), y (0) = 0, y^{^{'}} (0) = 0

limit as x approaches 2+of (|x|)/(2-x)

x \to 2 + lim (\frac{∣ x ∣}{2 - x})

integral of 30x+sin(x)+C

\int 30 x + sin (x) + C d x