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

解答

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

\frac{x ^{2}}{2} + x - \frac{4}{3} arctan (\frac{1}{3} (2 x - 1)) - ln 4 x^{2} - 4 x + 4 + 23 arctan (\frac{1}{3} (2 x - 1)) + C

求解步骤

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

乘开

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

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

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

\int \frac{3}{x ^{2} - x + 1} d x = 23 arctan (\frac{1}{3} (2 x - 1))

= \frac{x ^{2}}{2} + x - \frac{2}{3} arctan (\frac{2 x - 1}{3}) - (ln 4 x^{2} - 4 x + 4 + \frac{2}{3} arctan (\frac{2 x - 1}{3})) + 23 arctan (\frac{1}{3} (2 x - 1))

化简

\frac{x ^{2}}{2} + x - \frac{2}{3} arctan (\frac{2 x - 1}{3}) - (ln 4 x^{2} - 4 x + 4 + \frac{2}{3} arctan (\frac{2 x - 1}{3})) + 23 arctan (\frac{1}{3} (2 x - 1)) : \frac{x ^{2}}{2} + x - \frac{4}{3} arctan (\frac{1}{3} (2 x - 1)) - ln 4 x^{2} - 4 x + 4 + 23 arctan (\frac{1}{3} (2 x - 1))

= \frac{x ^{2}}{2} + x - \frac{4}{3} arctan (\frac{1}{3} (2 x - 1)) - ln 4 x^{2} - 4 x + 4 + 23 arctan (\frac{1}{3} (2 x - 1))

解答补常数

= \frac{x ^{2}}{2} + x - \frac{4}{3} arctan (\frac{1}{3} (2 x - 1)) - ln 4 x^{2} - 4 x + 4 + 23 arctan (\frac{1}{3} (2 x - 1)) + C

\frac{\partial}{\partial x} ((1 - x) + 10 (y - x^{2})^{2})

\int_{1}^{\infty} \frac{1}{1 + x ^{2}} d x

\frac{d}{d x} (x - 2 - 0.5 sin (x))

\frac{d}{d s} (s t)

d er i v a t i v e f (x) = - 4 sin (x)