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受欢迎的微积分 >

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

解答

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

解答

- \frac{1}{2} ln ∣ x + 2 ∣ + \frac{1}{2} ln ∣ x ∣ + C

求解步骤

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

对

2 x + x^{2}

配方:

(x + 1)^{2} - 1

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

使用换元积分法

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

因式分解

u^{2} - 1 : - (- u^{2} + 1)

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

提出常数:

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

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

使用积分常见定则:

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

= - (\frac{ln ∣ u + 1 ∣}{2} - \frac{ln ∣ u - 1 ∣}{2})

u = x + 1

代回

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

化简

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

= - \frac{1}{2} ln ∣ x + 2 ∣ + \frac{1}{2} ln ∣ x ∣

解答补常数

= - \frac{1}{2} ln ∣ x + 2 ∣ + \frac{1}{2} ln ∣ x ∣ + C

作图

流行的例子

integral of 2/(x^3+1)

\int \frac{2}{x ^{3} + 1} d x

integral of e^{3x}x+e^x

\int e^{3 x} x + e^{x} d x

integral of x/(27x^2-1)

\int \frac{x}{27 x ^{2} - 1} d x

limit as x approaches 0+of-x+3

x \to 0 + lim (- x + 3)

(d^2y)/(dx^2)+25y=0

\frac{d ^{2} y}{d x ^{2}} + 25 y = 0