integral of (2x^2-5x-2)/((x-2)^2(x-1))

解答

\int \frac{2 x ^{2} - 5 x - 2}{( x - 2 ) ^{2} ( x - 1 )} d x

7 ln ∣ x - 2 ∣ + \frac{4}{x - 2} - 5 ln ∣ x - 1 ∣ + C

求解步骤

\int \frac{2 x ^{2} - 5 x - 2}{( x - 2 ) ^{2} ( x - 1 )} d x

将

\frac{2 x ^{2} - 5 x - 2}{( x - 2 ) ^{2} ( x - 1 )}

用部份分式展开:

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

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

\int \frac{7}{x - 2} d x = 7 ln ∣ x - 2 ∣

\int \frac{4}{( x - 2 ) ^{2}} d x = - \frac{4}{x - 2}

\int \frac{5}{x - 1} d x = 5 ln ∣ x - 1 ∣

= 7 ln ∣ x - 2 ∣ - (- \frac{4}{x - 2}) - 5 ln ∣ x - 1 ∣

化简

= 7 ln ∣ x - 2 ∣ + \frac{4}{x - 2} - 5 ln ∣ x - 1 ∣

解答补常数

= 7 ln ∣ x - 2 ∣ + \frac{4}{x - 2} - 5 ln ∣ x - 1 ∣ + C

t an g e n t f (x) = 5 x + 30, at x = - 1

x \to - 1 + lim (∣ x ∣ - 3)

\int_{- 1}^{1} (x^{100}) d x

\frac{d}{d x} (3 x^{2})

\frac{\partial}{\partial x} (\frac{x y}{x ^{2} + z ^{2}})