integral of (4x^2-x+25)/(x^3+25x)

解答

\int \frac{4 x ^{2} - x + 25}{x ^{3} + 25 x} d x

- \frac{1}{5} arctan (\frac{x}{5}) + \frac{3}{2} ln x^{2} + 25 + ln ∣ x ∣ + C

求解步骤

\int \frac{4 x ^{2} - x + 25}{x ^{3} + 25 x} d x

乘开

\frac{4 x ^{2} - x + 25}{x ^{3} + 25 x} : \frac{4 x ^{2}}{x ^{3} + 25 x} - \frac{x}{x ^{3} + 25 x} + \frac{25}{x ^{3} + 25 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{4 x ^{2}}{x ^{3} + 25 x} d x - \int \frac{x}{x ^{3} + 25 x} d x + \int \frac{25}{x ^{3} + 25 x} d x

\int \frac{4 x ^{2}}{x ^{3} + 25 x} d x = 2 ln x^{2} + 25

\int \frac{x}{x ^{3} + 25 x} d x = \frac{1}{5} arctan (\frac{x}{5})

\int \frac{25}{x ^{3} + 25 x} d x = ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 25

= 2 ln x^{2} + 25 - \frac{1}{5} arctan (\frac{x}{5}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 25

化简

2 ln x^{2} + 25 - \frac{1}{5} arctan (\frac{x}{5}) + ln ∣ x ∣ - \frac{1}{2} ln x^{2} + 25 : - \frac{1}{5} arctan (\frac{x}{5}) + \frac{3}{2} ln x^{2} + 25 + ln ∣ x ∣

= - \frac{1}{5} arctan (\frac{x}{5}) + \frac{3}{2} ln x^{2} + 25 + ln ∣ x ∣

解答补常数

= - \frac{1}{5} arctan (\frac{x}{5}) + \frac{3}{2} ln x^{2} + 25 + ln ∣ x ∣ + C

d er i v a t i v e 1 + \frac{1}{x}

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

t an g e n t f (x) = 9 x^{2} + 94, at x = 6

\frac{\partial}{\partial x} (\frac{3}{x} - 1 - y)

a re a 4 x + 8, x^{2} + 19 x + 62, [- 9, - 6]