integral of (x^2-x+30)/(x^3+5x)

解答

\int \frac{x ^{2} - x + 30}{x ^{3} + 5 x} d x

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

求解步骤

\int \frac{x ^{2} - x + 30}{x ^{3} + 5 x} d x

将

\frac{x ^{2} - x + 30}{x ^{3} + 5 x}

用部份分式展开:

\frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 5}

= \int \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 5} d x

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{6}{x} d x + \int \frac{- 5 x - 1}{x ^{2} + 5} d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

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

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 5 - \frac{1}{5} arctan (\frac{x}{5})

解答补常数

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 5 - \frac{1}{5} arctan (\frac{x}{5}) + C

\frac{\partial}{\partial x} (cos (w + x) sin (y - z))

\frac{d}{d t} (3 t - 2)

\int_{0}^{1} \frac{1}{( x + 1 ) ( x + 2 )} d x

t an g e n t f (x) = 17 tan (x), at x = \frac{π}{4}

d er i v a t i v e 5 x^{4} 25 - x^{2}