integral of (3x^2-5x-17)/((x+3)(x-2)^2)

解答

\int \frac{3 x ^{2} - 5 x - 17}{( x + 3 ) ( x - 2 ) ^{2}} d x

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

求解步骤

\int \frac{3 x ^{2} - 5 x - 17}{( x + 3 ) ( x - 2 ) ^{2}} d x

使用换元积分法

= \int \frac{3 u ^{2} - 23 u + 25}{u ( u - 5 ) ^{2}} d u

乘开

\frac{3 u ^{2} - 23 u + 25}{u ( u - 5 ) ^{2}} : \frac{3 u ^{2}}{u ( u - 5 ) ^{2}} - \frac{23 u}{u ( u - 5 ) ^{2}} + \frac{25}{u ( u - 5 ) ^{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{3 u ^{2}}{u ( u - 5 ) ^{2}} d u - \int \frac{23 u}{u ( u - 5 ) ^{2}} d u + \int \frac{25}{u ( u - 5 ) ^{2}} d u

\int \frac{3 u ^{2}}{u ( u - 5 ) ^{2}} d u = 3 (ln ∣ u - 5 ∣ - \frac{5}{u - 5})

\int \frac{23 u}{u ( u - 5 ) ^{2}} d u = - \frac{23}{u - 5}

\int \frac{25}{u ( u - 5 ) ^{2}} d u = ln ∣ u ∣ - ln ∣ u - 5 ∣ - \frac{5}{u - 5}

= 3 (ln ∣ u - 5 ∣ - \frac{5}{u - 5}) - (- \frac{23}{u - 5}) + ln ∣ u ∣ - ln ∣ u - 5 ∣ - \frac{5}{u - 5}

u = x + 3

代回

= 3 (ln ∣ x + 3 - 5 ∣ - \frac{5}{x + 3 - 5}) - (- \frac{23}{x + 3 - 5}) + ln ∣ x + 3 ∣ - ln ∣ x + 3 - 5 ∣ - \frac{5}{x + 3 - 5}

化简

3 (ln ∣ x + 3 - 5 ∣ - \frac{5}{x + 3 - 5}) - (- \frac{23}{x + 3 - 5}) + ln ∣ x + 3 ∣ - ln ∣ x + 3 - 5 ∣ - \frac{5}{x + 3 - 5} : 2 ln ∣ x - 2 ∣ + \frac{3}{x - 2} + ln ∣ x + 3 ∣

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

解答补常数

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

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

\int (2 x - 4) (x + 2) d x

a re a x - 4, x = 8

2 (y - 4 x^{2}) d x + x d y = 0

d er i v a t i v e 8 x e^{x} csc (x)