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受欢迎的微积分 >

integral of 1/(sqrt(x^2-x))

解答

\int \frac{1}{x ^{2} - x} d x

解答

2 ln x + x - 1 + C

求解步骤

\int \frac{1}{x ^{2} - x} d x

分解

\frac{1}{x ^{2} - x}

= \int \frac{1}{x ( x - 1 )} d x

使用换元积分法

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

使用指数法则:

(a \cdot b)^{n} = a^{n} \cdot b^{n}

(u + 1) u = (u + 1) u

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

使用换元积分法

= \int \frac{2}{v ^{2} + 1} d v

提出常数:

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

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

使用积分常见定则:

\int \frac{1}{v ^{2} + 1} d v = ln v^{2} + 1 + v

= 2 ln v^{2} + 1 + v

代回

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

(x - 1)^{2} + 1 = x

= 2 ln x + x - 1

解答补常数

= 2 ln x + x - 1 + C

作图

流行的例子

(dy)/(dx)=(((2y+3))/((4x+5)))^2

\frac{d y}{d x} = (\frac{( 2 y + 3 )}{( 4 x + 5 )})^{2}

面积 cos(6x),0

a re a cos (6 x), 0

y^{''}+4sqrt(3)y^'+12y=0,y(0)=1,y^'(0)=0

y^{^{''}} + 43 y^{^{'}} + 12 y = 0, y (0) = 1, y^{^{'}} (0) = 0

sum from n=1 to infinity of 2/(n+1)

n = 1 \sum \infty \frac{2}{n + 1}

integral from 0 to pi of x+sin(x)

\int_{0}^{π} x + sin (x) d x