integral of (11-2x)/(x^2+9x+14)

解答

\int \frac{11 - 2 x}{x ^{2} + 9 x + 14} d x

\frac{11}{5} ln ∣ x + 2 ∣ - \frac{11}{5} ln ∣ x + 7 ∣ - ln 4 x^{2} + 36 x + 56 - \frac{9}{5} ln \frac{2}{5} x + \frac{14}{5} + \frac{9}{5} ln \frac{2}{5} x + \frac{4}{5} + C

求解步骤

\int \frac{11 - 2 x}{x ^{2} + 9 x + 14} d x

乘开

\frac{11 - 2 x}{x ^{2} + 9 x + 14} : \frac{11}{x ^{2} + 9 x + 14} - \frac{2 x}{x ^{2} + 9 x + 14}

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{11}{x ^{2} + 9 x + 14} d x - \int \frac{2 x}{x ^{2} + 9 x + 14} d x

\int \frac{11}{x ^{2} + 9 x + 14} d x = 11 (\frac{1}{5} ln ∣ x + 2 ∣ - \frac{1}{5} ln ∣ x + 7 ∣)

\int \frac{2 x}{x ^{2} + 9 x + 14} d x = ln 4 x^{2} + 36 x + 56 + \frac{9}{5} (ln \frac{2}{5} x + \frac{14}{5} - ln \frac{2}{5} x + \frac{4}{5})

= 11 (\frac{1}{5} ln ∣ x + 2 ∣ - \frac{1}{5} ln ∣ x + 7 ∣) - (ln 4 x^{2} + 36 x + 56 + \frac{9}{5} (ln \frac{2}{5} x + \frac{14}{5} - ln \frac{2}{5} x + \frac{4}{5}))

化简

11 (\frac{1}{5} ln ∣ x + 2 ∣ - \frac{1}{5} ln ∣ x + 7 ∣) - (ln 4 x^{2} + 36 x + 56 + \frac{9}{5} (ln \frac{2}{5} x + \frac{14}{5} - ln \frac{2}{5} x + \frac{4}{5})) : \frac{11}{5} ln ∣ x + 2 ∣ - \frac{11}{5} ln ∣ x + 7 ∣ - ln 4 x^{2} + 36 x + 56 - \frac{9}{5} ln \frac{2}{5} x + \frac{14}{5} + \frac{9}{5} ln \frac{2}{5} x + \frac{4}{5}

= \frac{11}{5} ln ∣ x + 2 ∣ - \frac{11}{5} ln ∣ x + 7 ∣ - ln 4 x^{2} + 36 x + 56 - \frac{9}{5} ln \frac{2}{5} x + \frac{14}{5} + \frac{9}{5} ln \frac{2}{5} x + \frac{4}{5}

解答补常数

= \frac{11}{5} ln ∣ x + 2 ∣ - \frac{11}{5} ln ∣ x + 7 ∣ - ln 4 x^{2} + 36 x + 56 - \frac{9}{5} ln \frac{2}{5} x + \frac{14}{5} + \frac{9}{5} ln \frac{2}{5} x + \frac{4}{5} + C

x \to 3 lim ((x^{2} - 9)^{x - 3})

x \to \infty lim (\frac{x}{9})

\frac{d y}{d x} = \frac{1}{2 y + 1}

\int \frac{1}{136} x^{3} d x

d er i v a t i v e y = \frac{- 9 e ^{4 x}}{8 x + 3}