integral of (5x^4+7x^2+x+2)/(x(x^2+1)^2)

解答

\int \frac{5 x ^{4} + 7 x ^{2} + x + 2}{x ( x ^{2} + 1 ) ^{2}} d x

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

求解步骤

\int \frac{5 x ^{4} + 7 x ^{2} + x + 2}{x ( x ^{2} + 1 ) ^{2}} d x

将

\frac{5 x ^{4} + 7 x ^{2} + x + 2}{x ( x ^{2} + 1 ) ^{2}}

用部份分式展开:

\frac{2}{x} + \frac{3 x}{x ^{2} + 1} + \frac{1}{( x ^{2} + 1 ) ^{2}}

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

\int \frac{2}{x} d x = 2 ln ∣ x ∣

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

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

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

解答补常数

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

\int \frac{1}{x ln ( x ^{3} )} d x

\frac{\partial}{\partial x} (e^{x y} cos (y))

\frac{d y}{d x} = \frac{3 y ^{3}}{x ^{2}}, y (3) = 1

(2 sin (x))^{^{'}}

s l o p e (- 29, - 40), (- 28, 3)