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

解答

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

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

求解步骤

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

乘开

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

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

\int \frac{2 x}{x ^{2} + 4} d x = ln x^{2} + 4

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

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

化简

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

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

解答补常数

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

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