integral of (x^2+3x-2)/(x^2-1)

解答

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

x + \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{3}{2} ln x^{2} - 1 + C

求解步骤

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

乘开

\frac{x ^{2} + 3 x - 2}{x ^{2} - 1} : \frac{x ^{2}}{x ^{2} - 1} + \frac{3 x}{x ^{2} - 1} - \frac{2}{x ^{2} - 1}

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

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

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

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

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

化简

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

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

解答补常数

= x + \frac{1}{2} ln ∣ x + 1 ∣ - \frac{1}{2} ln ∣ x - 1 ∣ + \frac{3}{2} ln x^{2} - 1 + C

\frac{d}{d x} (\frac{1}{2 sin ( x )})

t an g e n t 2 sin (x)

\frac{\partial}{\partial m} (- m g (m) sin (a))

x \to 3 lim (\frac{27 - x}{x - 3})

\int_{0}^{π} π (sin^{2} (x)) d x