integral of (x+7)/(x^2+2x)

解答

\int \frac{x + 7}{x ^{2} + 2 x} d x

- \frac{5}{2} ln ∣ x + 2 ∣ + \frac{7}{2} ln ∣ x ∣ + C

求解步骤

\int \frac{x + 7}{x ^{2} + 2 x} d x

乘开

\frac{x + 7}{x ^{2} + 2 x} : \frac{x}{x ^{2} + 2 x} + \frac{7}{x ^{2} + 2 x}

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{x}{x ^{2} + 2 x} d x + \int \frac{7}{x ^{2} + 2 x} d x

\int \frac{x}{x ^{2} + 2 x} d x = ln ∣ x + 2 ∣

\int \frac{7}{x ^{2} + 2 x} d x = - 7 (\frac{1}{2} ln ∣ x + 2 ∣ - \frac{1}{2} ln ∣ x ∣)

= ln ∣ x + 2 ∣ - 7 (\frac{1}{2} ln ∣ x + 2 ∣ - \frac{1}{2} ln ∣ x ∣)

化简

ln ∣ x + 2 ∣ - 7 (\frac{1}{2} ln ∣ x + 2 ∣ - \frac{1}{2} ln ∣ x ∣) : - \frac{5}{2} ln ∣ x + 2 ∣ + \frac{7}{2} ln ∣ x ∣

= - \frac{5}{2} ln ∣ x + 2 ∣ + \frac{7}{2} ln ∣ x ∣

解答补常数

= - \frac{5}{2} ln ∣ x + 2 ∣ + \frac{7}{2} ln ∣ x ∣ + C

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