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受欢迎的微积分 >

integral of 1/(x^{1/2)-4}

解答

\int \frac{1}{x ^{\frac{1}{2}} - 4} d x

解答

2 (x^{\frac{1}{2}} - 4 + 4 ln x^{\frac{1}{2}} - 4) + C

求解步骤

\int \frac{1}{x ^{\frac{1}{2}} - 4} d x

使用换元积分法

= \int \frac{2 u}{u - 4} 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}{u - 4} d u

使用换元积分法

= 2 \cdot \int \frac{v + 4}{v} d v

乘开

\frac{v + 4}{v} : 1 + \frac{4}{v}

= 2 \cdot \int 1 + \frac{4}{v} d v

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 1 d v + \int \frac{4}{v} d v)

\int 1 d v = v

\int \frac{4}{v} d v = 4 ln ∣ v ∣

= 2 (v + 4 ln ∣ v ∣)

代回

= 2 (x^{\frac{1}{2}} - 4 + 4 ln x^{\frac{1}{2}} - 4)

解答补常数

= 2 (x^{\frac{1}{2}} - 4 + 4 ln x^{\frac{1}{2}} - 4) + C

作图

流行的例子

integral of x^2sqrt(5-2x^3)

\int x^{2} 5 - 2 x^{3} d x

integral of (x^{1.8}+11x^{4.5})

\int (x^{1.8} + 11 x^{4.5}) d x

(\partial)/(\partial y)(7e^xln(y))

\frac{\partial}{\partial y} (7 e^{x} ln (y))

limit as x approaches 6 of x(x-3)

x \to 6 lim (x (x - 3))

y^{''}-18y^'+130y=0,y(0)=0,y(pi/2)=1

y^{^{''}} - 18 y^{^{'}} + 130 y = 0, y (0) = 0, y (\frac{π}{2}) = 1