integral of (x+8)/(x^3+13x^2+35x-49)

解答

\int \frac{x + 8}{x ^{3} + 13 x ^{2} + 35 x - 49} d x

\frac{9}{64} ln ∣ x - 1 ∣ - \frac{9}{64} ln ∣ x + 7 ∣ - \frac{7}{8 ( x + 7 )} + \frac{1}{x + 7} + C

求解步骤

\int \frac{x + 8}{x ^{3} + 13 x ^{2} + 35 x - 49} d x

乘开

\frac{x + 8}{x ^{3} + 13 x ^{2} + 35 x - 49} : \frac{x}{x ^{3} + 13 x ^{2} + 35 x - 49} + \frac{8}{x ^{3} + 13 x ^{2} + 35 x - 49}

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 ^{3} + 13 x ^{2} + 35 x - 49} d x + \int \frac{8}{x ^{3} + 13 x ^{2} + 35 x - 49} d x

\int \frac{x}{x ^{3} + 13 x ^{2} + 35 x - 49} d x = \frac{1}{64} ln ∣ x - 1 ∣ - \frac{1}{64} ln ∣ x + 7 ∣ - \frac{7}{8 ( x + 7 )}

\int \frac{8}{x ^{3} + 13 x ^{2} + 35 x - 49} d x = \frac{1}{8} ln ∣ x - 1 ∣ - \frac{1}{8} ln ∣ x + 7 ∣ + \frac{1}{x + 7}

= \frac{1}{64} ln ∣ x - 1 ∣ - \frac{1}{64} ln ∣ x + 7 ∣ - \frac{7}{8 ( x + 7 )} + \frac{1}{8} ln ∣ x - 1 ∣ - \frac{1}{8} ln ∣ x + 7 ∣ + \frac{1}{x + 7}

化简

\frac{1}{64} ln ∣ x - 1 ∣ - \frac{1}{64} ln ∣ x + 7 ∣ - \frac{7}{8 ( x + 7 )} + \frac{1}{8} ln ∣ x - 1 ∣ - \frac{1}{8} ln ∣ x + 7 ∣ + \frac{1}{x + 7} : \frac{9}{64} ln ∣ x - 1 ∣ - \frac{9}{64} ln ∣ x + 7 ∣ - \frac{7}{8 ( x + 7 )} + \frac{1}{x + 7}

= \frac{9}{64} ln ∣ x - 1 ∣ - \frac{9}{64} ln ∣ x + 7 ∣ - \frac{7}{8 ( x + 7 )} + \frac{1}{x + 7}

解答补常数

= \frac{9}{64} ln ∣ x - 1 ∣ - \frac{9}{64} ln ∣ x + 7 ∣ - \frac{7}{8 ( x + 7 )} + \frac{1}{x + 7} + C

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