integral of ((5x^2+10x-30))/(x^2+x-6)

解答

\int \frac{( 5 x ^{2} + 10 x - 30 )}{x ^{2} + x - 6} d x

5 x + 4 ln ∣ x - 2 ∣ - 9 ln ∣ x + 3 ∣ + 5 ln 4 x^{2} + 4 x - 24 + 7 ln \frac{2}{5} x + \frac{6}{5} - 7 ln \frac{2}{5} x - \frac{4}{5} + C

求解步骤

\int \frac{5 x ^{2} + 10 x - 30}{x ^{2} + x - 6} d x

乘开

\frac{5 x ^{2} + 10 x - 30}{x ^{2} + x - 6} : \frac{5 x ^{2}}{x ^{2} + x - 6} + \frac{10 x}{x ^{2} + x - 6} - \frac{30}{x ^{2} + x - 6}

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{5 x ^{2}}{x ^{2} + x - 6} d x + \int \frac{10 x}{x ^{2} + x - 6} d x - \int \frac{30}{x ^{2} + x - 6} d x

\int \frac{5 x ^{2}}{x ^{2} + x - 6} d x = 5 x + 4 ln ∣ x - 2 ∣ - 9 ln ∣ x + 3 ∣

\int \frac{10 x}{x ^{2} + x - 6} d x = 5 ln 4 x^{2} + 4 x - 24 + ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x

\int \frac{30}{x ^{2} + x - 6} d x = - 12 (\frac{1}{2} ln \frac{6}{5} + \frac{2}{5} x - \frac{1}{2} ln - \frac{4}{5} + \frac{2}{5} x)

= 5 x + 4 ln ∣ x - 2 ∣ - 9 ln ∣ x + 3 ∣ + 5 ln 4 x^{2} + 4 x - 24 + ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x - (- 12 (\frac{1}{2} ln \frac{6}{5} + \frac{2}{5} x - \frac{1}{2} ln - \frac{4}{5} + \frac{2}{5} x))

化简

5 x + 4 ln ∣ x - 2 ∣ - 9 ln ∣ x + 3 ∣ + 5 ln 4 x^{2} + 4 x - 24 + ln \frac{6}{5} + \frac{2}{5} x - ln - \frac{4}{5} + \frac{2}{5} x - (- 12 (\frac{1}{2} ln \frac{6}{5} + \frac{2}{5} x - \frac{1}{2} ln - \frac{4}{5} + \frac{2}{5} x)) : 5 x + 4 ln ∣ x - 2 ∣ - 9 ln ∣ x + 3 ∣ + 5 ln 4 x^{2} + 4 x - 24 + 7 ln \frac{2}{5} x + \frac{6}{5} - 7 ln \frac{2}{5} x - \frac{4}{5}

= 5 x + 4 ln ∣ x - 2 ∣ - 9 ln ∣ x + 3 ∣ + 5 ln 4 x^{2} + 4 x - 24 + 7 ln \frac{2}{5} x + \frac{6}{5} - 7 ln \frac{2}{5} x - \frac{4}{5}

解答补常数

= 5 x + 4 ln ∣ x - 2 ∣ - 9 ln ∣ x + 3 ∣ + 5 ln 4 x^{2} + 4 x - 24 + 7 ln \frac{2}{5} x + \frac{6}{5} - 7 ln \frac{2}{5} x - \frac{4}{5} + C

\int \frac{16 x}{x ^{2} + 1} d x

\int \frac{ln ( 2 x )}{x} d x

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