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

解答

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

\frac{x ^{3}}{3} - 5 x + 55 arctan (\frac{x}{5}) + \frac{1}{2} (x^{2} + 5 - 5 ln x^{2} + 5) - \frac{2}{5} arctan (\frac{x}{5}) + C

求解步骤

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

乘开

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

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

\int \frac{x ^{4}}{x ^{2} + 5} d x = \frac{x ^{3}}{3} - 5 x + 55 arctan (\frac{x}{5})

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

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

= \frac{x ^{3}}{3} - 5 x + 55 arctan (\frac{x}{5}) + \frac{1}{2} (x^{2} + 5 - 5 ln x^{2} + 5) - \frac{2}{5} arctan (\frac{x}{5})

解答补常数

= \frac{x ^{3}}{3} - 5 x + 55 arctan (\frac{x}{5}) + \frac{1}{2} (x^{2} + 5 - 5 ln x^{2} + 5) - \frac{2}{5} arctan (\frac{x}{5}) + C

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