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

解答

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

\frac{4}{5} arctan (x) + \frac{3}{5} ln ∣ x - 2 ∣ + \frac{1}{5} ln x^{2} + 1 + C

求解步骤

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

乘开

\frac{x ^{2} - 1}{( x ^{2} + 1 ) ( x - 2 )} : \frac{x ^{2}}{( x ^{2} + 1 ) ( x - 2 )} - \frac{1}{( x ^{2} + 1 ) ( x - 2 )}

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

\int \frac{x ^{2}}{( x ^{2} + 1 ) ( x - 2 )} d x = \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 2 arctan (x)) + \frac{4}{5} ln ∣ x - 2 ∣

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

= \frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 2 arctan (x)) + \frac{4}{5} ln ∣ x - 2 ∣ - (\frac{1}{5} (- \frac{1}{2} ln x^{2} + 1 - 2 arctan (x)) + \frac{1}{5} ln ∣ x - 2 ∣)

化简

\frac{1}{5} (\frac{1}{2} ln x^{2} + 1 + 2 arctan (x)) + \frac{4}{5} ln ∣ x - 2 ∣ - (\frac{1}{5} (- \frac{1}{2} ln x^{2} + 1 - 2 arctan (x)) + \frac{1}{5} ln ∣ x - 2 ∣) : \frac{4}{5} arctan (x) + \frac{3}{5} ln ∣ x - 2 ∣ + \frac{1}{5} ln x^{2} + 1

= \frac{4}{5} arctan (x) + \frac{3}{5} ln ∣ x - 2 ∣ + \frac{1}{5} ln x^{2} + 1

解答补常数

= \frac{4}{5} arctan (x) + \frac{3}{5} ln ∣ x - 2 ∣ + \frac{1}{5} ln x^{2} + 1 + C

\frac{\partial}{\partial x} (\frac{5 x - 3 y}{5 x + 3 y})

\int x^{2} cos (x^{3}) d x

x \to 1 lim (- x^{2} + 9 x - 7)

(1 + e^{x}) \frac{d y}{d x} + e^{x} y = 0

\int \frac{3 x}{x + 2} d x