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受欢迎的微积分 >

integral of (x+4)/(x^2+5x-6)

解答

\int \frac{x + 4}{x ^{2} + 5 x - 6} d x

解答

ln ∣ x + 6 ∣ + 5 (\frac{1}{7} ln ∣ x - 1 ∣ - \frac{1}{7} ln ∣ x + 6 ∣) + C

求解步骤

\int \frac{x + 4}{x ^{2} + 5 x - 6} d x

分解

x^{2} + 5 x - 6 : (x - 1) (x + 6)

= \int \frac{x + 4}{( x - 1 ) ( x + 6 )} d x

使用换元积分法

= \int \frac{u + 5}{u ( u + 7 )} d u

乘开

\frac{u + 5}{u ( u + 7 )} : \frac{u}{u ( u + 7 )} + \frac{5}{u ( u + 7 )}

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 \frac{u}{u ( u + 7 )} d u + \int \frac{5}{u ( u + 7 )} d u

\int \frac{u}{u ( u + 7 )} d u = ln ∣ u + 7 ∣

\int \frac{5}{u ( u + 7 )} d u = 5 (\frac{1}{7} ln ∣ u ∣ - \frac{1}{7} ln ∣ u + 7 ∣)

= ln ∣ u + 7 ∣ + 5 (\frac{1}{7} ln ∣ u ∣ - \frac{1}{7} ln ∣ u + 7 ∣)

u = x - 1

代回

= ln ∣ x - 1 + 7 ∣ + 5 (\frac{1}{7} ln ∣ x - 1 ∣ - \frac{1}{7} ln ∣ x - 1 + 7 ∣)

化简

ln ∣ x - 1 + 7 ∣ + 5 (\frac{1}{7} ln ∣ x - 1 ∣ - \frac{1}{7} ln ∣ x - 1 + 7 ∣) : ln ∣ x + 6 ∣ + 5 (\frac{1}{7} ln ∣ x - 1 ∣ - \frac{1}{7} ln ∣ x + 6 ∣)

= ln ∣ x + 6 ∣ + 5 (\frac{1}{7} ln ∣ x - 1 ∣ - \frac{1}{7} ln ∣ x + 6 ∣)

解答补常数

= ln ∣ x + 6 ∣ + 5 (\frac{1}{7} ln ∣ x - 1 ∣ - \frac{1}{7} ln ∣ x + 6 ∣) + C

作图

流行的例子

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\frac{d y}{d x} = 6 x y

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