integral of (x^3-x^2+x-2)/(x^2+1)

解答

\int \frac{x ^{3} - x ^{2} + x - 2}{x ^{2} + 1} d x

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

求解步骤

\int \frac{x ^{3} - x ^{2} + x - 2}{x ^{2} + 1} d x

乘开

\frac{x ^{3} - x ^{2} + x - 2}{x ^{2} + 1} : \frac{x ^{3}}{x ^{2} + 1} - \frac{x ^{2}}{x ^{2} + 1} + \frac{x}{x ^{2} + 1} - \frac{2}{x ^{2} + 1}

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

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

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

\int \frac{x}{x ^{2} + 1} d x = \frac{1}{2} ln x^{2} + 1

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

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

化简

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

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

解答补常数

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

\int \frac{x ^{2} + 1}{( 2 x - 3 ) ^{2}} d x

\int sin (e^{3 x}) d x

\int \frac{1}{( e ^{x} - e ^{- x} )} d x

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