integral of (x^3+x^2+x-5)/(x^2+2x+4)

解答

\int \frac{x ^{3} + x ^{2} + x - 5}{x ^{2} + 2 x + 4} d x

- 2 x - \frac{1}{2} ln x^{2} + 2 x + 4 + \frac{8}{3} arctan (\frac{1}{3} (x + 1)) - \frac{8}{3} arctan (\frac{1}{3} (x + 1)) + \frac{1}{2} (x + 1)^{2} - 3 + C

求解步骤

\int \frac{x ^{3} + x ^{2} + x - 5}{x ^{2} + 2 x + 4} d x

乘开

\frac{x ^{3} + x ^{2} + x - 5}{x ^{2} + 2 x + 4} : \frac{x ^{3}}{x ^{2} + 2 x + 4} + \frac{x ^{2}}{x ^{2} + 2 x + 4} + \frac{x}{x ^{2} + 2 x + 4} - \frac{5}{x ^{2} + 2 x + 4}

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 ^{3}}{x ^{2} + 2 x + 4} d x + \int \frac{x ^{2}}{x ^{2} + 2 x + 4} d x + \int \frac{x}{x ^{2} + 2 x + 4} d x - \int \frac{5}{x ^{2} + 2 x + 4} d x

\int \frac{x ^{3}}{x ^{2} + 2 x + 4} d x = \frac{( x + 1 ) ^{2}}{2} - 3 (x + 1) + \frac{8}{3} arctan (\frac{x + 1}{3})

\int \frac{x ^{2}}{x ^{2} + 2 x + 4} d x = - ln x^{2} + 2 x + 4 - \frac{2}{3} arctan (\frac{1}{3} (x + 1)) + x

\int \frac{x}{x ^{2} + 2 x + 4} d x = \frac{1}{2} ln x^{2} + 2 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (x + 1))

\int \frac{5}{x ^{2} + 2 x + 4} d x = \frac{5}{3} arctan (\frac{1}{3} (x + 1))

= \frac{( x + 1 ) ^{2}}{2} - 3 (x + 1) + \frac{8}{3} arctan (\frac{x + 1}{3}) - ln x^{2} + 2 x + 4 - \frac{2}{3} arctan (\frac{1}{3} (x + 1)) + x + \frac{1}{2} ln x^{2} + 2 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (x + 1)) - \frac{5}{3} arctan (\frac{1}{3} (x + 1))

化简

\frac{( x + 1 ) ^{2}}{2} - 3 (x + 1) + \frac{8}{3} arctan (\frac{x + 1}{3}) - ln x^{2} + 2 x + 4 - \frac{2}{3} arctan (\frac{1}{3} (x + 1)) + x + \frac{1}{2} ln x^{2} + 2 x + 4 - \frac{1}{3} arctan (\frac{1}{3} (x + 1)) - \frac{5}{3} arctan (\frac{1}{3} (x + 1)) : - 2 x - \frac{1}{2} ln x^{2} + 2 x + 4 + \frac{8}{3} arctan (\frac{1}{3} (x + 1)) - \frac{8}{3} arctan (\frac{1}{3} (x + 1)) + \frac{1}{2} (x + 1)^{2} - 3

= - 2 x - \frac{1}{2} ln x^{2} + 2 x + 4 + \frac{8}{3} arctan (\frac{1}{3} (x + 1)) - \frac{8}{3} arctan (\frac{1}{3} (x + 1)) + \frac{1}{2} (x + 1)^{2} - 3

解答补常数

= - 2 x - \frac{1}{2} ln x^{2} + 2 x + 4 + \frac{8}{3} arctan (\frac{1}{3} (x + 1)) - \frac{8}{3} arctan (\frac{1}{3} (x + 1)) + \frac{1}{2} (x + 1)^{2} - 3 + C

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