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受欢迎的微积分 >

integral of (x+4)/(x^2+6x-16)

解答

\int \frac{x + 4}{x ^{2} + 6 x - 16} d x

解答

ln ∣ x + 8 ∣ + 6 (\frac{1}{10} ln ∣ x - 2 ∣ - \frac{1}{10} ln ∣ x + 8 ∣) + C

求解步骤

\int \frac{x + 4}{x ^{2} + 6 x - 16} d x

分解

x^{2} + 6 x - 16 : (x - 2) (x + 8)

= \int \frac{x + 4}{( x - 2 ) ( x + 8 )} d x

使用换元积分法

= \int \frac{u + 6}{u ( u + 10 )} d u

乘开

\frac{u + 6}{u ( u + 10 )} : \frac{u}{u ( u + 10 )} + \frac{6}{u ( u + 10 )}

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 + 10 )} d u + \int \frac{6}{u ( u + 10 )} d u

\int \frac{u}{u ( u + 10 )} d u = ln ∣ u + 10 ∣

\int \frac{6}{u ( u + 10 )} d u = 6 (\frac{1}{10} ln ∣ u ∣ - \frac{1}{10} ln ∣ u + 10 ∣)

= ln ∣ u + 10 ∣ + 6 (\frac{1}{10} ln ∣ u ∣ - \frac{1}{10} ln ∣ u + 10 ∣)

u = x - 2

代回

= ln ∣ x - 2 + 10 ∣ + 6 (\frac{1}{10} ln ∣ x - 2 ∣ - \frac{1}{10} ln ∣ x - 2 + 10 ∣)

化简

ln ∣ x - 2 + 10 ∣ + 6 (\frac{1}{10} ln ∣ x - 2 ∣ - \frac{1}{10} ln ∣ x - 2 + 10 ∣) : ln ∣ x + 8 ∣ + 6 (\frac{1}{10} ln ∣ x - 2 ∣ - \frac{1}{10} ln ∣ x + 8 ∣)

= ln ∣ x + 8 ∣ + 6 (\frac{1}{10} ln ∣ x - 2 ∣ - \frac{1}{10} ln ∣ x + 8 ∣)

解答补常数

= ln ∣ x + 8 ∣ + 6 (\frac{1}{10} ln ∣ x - 2 ∣ - \frac{1}{10} ln ∣ x + 8 ∣) + C

作图

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