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

解答

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

x + 3 ln ∣ x - 1 ∣ - 2 ln ∣ x ∣ - 1 + C

求解步骤

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

乘开

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

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

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

= x - 1 + ln ∣ x - 1 ∣ + 2 (- ln ∣ x ∣ + ln ∣ x - 1 ∣)

化简

x - 1 + ln ∣ x - 1 ∣ + 2 (- ln ∣ x ∣ + ln ∣ x - 1 ∣) : x + 3 ln ∣ x - 1 ∣ - 2 ln ∣ x ∣ - 1

= x + 3 ln ∣ x - 1 ∣ - 2 ln ∣ x ∣ - 1

解答补常数

= x + 3 ln ∣ x - 1 ∣ - 2 ln ∣ x ∣ - 1 + C

\int 5 x + 4 y d x

\frac{d}{d x} ((ln (x + 4))^{2})

x \to 0 - lim (\frac{x ^{2}}{6} - \frac{7}{x})

x \to \infty lim (3 \cdot x^{3})

\frac{d x}{d t} = 1.2 x (1 - \frac{x}{4200})