integral of (x+2)/(x^2-2x+10)

解答

\int \frac{x + 2}{x ^{2} - 2 x + 10} d x

\frac{1}{2} ln x^{2} - 2 x + 10 + arctan (\frac{1}{3} (x - 1)) + C

求解步骤

\int \frac{x + 2}{x ^{2} - 2 x + 10} d x

乘开

\frac{x + 2}{x ^{2} - 2 x + 10} : \frac{x}{x ^{2} - 2 x + 10} + \frac{2}{x ^{2} - 2 x + 10}

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}{x ^{2} - 2 x + 10} d x + \int \frac{2}{x ^{2} - 2 x + 10} d x

\int \frac{x}{x ^{2} - 2 x + 10} d x = \frac{1}{2} ln x^{2} - 2 x + 10 + \frac{1}{3} arctan (\frac{1}{3} (x - 1))

\int \frac{2}{x ^{2} - 2 x + 10} d x = \frac{2}{3} arctan (\frac{1}{3} (x - 1))

= \frac{1}{2} ln x^{2} - 2 x + 10 + \frac{1}{3} arctan (\frac{1}{3} (x - 1)) + \frac{2}{3} arctan (\frac{1}{3} (x - 1))

化简

\frac{1}{2} ln x^{2} - 2 x + 10 + \frac{1}{3} arctan (\frac{1}{3} (x - 1)) + \frac{2}{3} arctan (\frac{1}{3} (x - 1)) : \frac{1}{2} ln x^{2} - 2 x + 10 + arctan (\frac{1}{3} (x - 1))

= \frac{1}{2} ln x^{2} - 2 x + 10 + arctan (\frac{1}{3} (x - 1))

解答补常数

= \frac{1}{2} ln x^{2} - 2 x + 10 + arctan (\frac{1}{3} (x - 1)) + C

d er i v a t i v e (\frac{4}{5} x)^{4}

\frac{\partial}{\partial x} (x - w)

x y^{^{'}} + 4 y = \frac{2 e ^{- 4 x}}{x ^{3}}

t an g e n t y = - 3 x^{2} + 10 x + 2, at x = 1

\frac{t d u}{d t} = t^{2} + 3 u, t > 0, H (6) = 180