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

解答

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

10 ln ∣ x - 3 ∣ - 9 ln ∣ x - 2 ∣ + \frac{5}{x - 2} + C

求解步骤

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

将

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

用部份分式展开:

\frac{10}{x - 3} - \frac{9}{x - 2} - \frac{5}{( x - 2 ) ^{2}}

= \int \frac{10}{x - 3} - \frac{9}{x - 2} - \frac{5}{( x - 2 ) ^{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{10}{x - 3} d x - \int \frac{9}{x - 2} d x - \int \frac{5}{( x - 2 ) ^{2}} d x

\int \frac{10}{x - 3} d x = 10 ln ∣ x - 3 ∣

\int \frac{9}{x - 2} d x = 9 ln ∣ x - 2 ∣

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

= 10 ln ∣ x - 3 ∣ - 9 ln ∣ x - 2 ∣ - (- \frac{5}{x - 2})

化简

= 10 ln ∣ x - 3 ∣ - 9 ln ∣ x - 2 ∣ + \frac{5}{x - 2}

解答补常数

= 10 ln ∣ x - 3 ∣ - 9 ln ∣ x - 2 ∣ + \frac{5}{x - 2} + C

im pl i c i t \frac{d y}{d x}, x y^{3} + 10 y^{11} = 9

a re a y^{2} - 3 x = 1, x - y = 3

\frac{d y}{d t} = - 100 y + 101 cos (t) + 99 sin (t)

t an g e n t y = 2 e^{x} - 3 x + 1, at x = 0

d er i v a t i v e 4 x^{2} - 12 x + 5