by using ostrogadski method solve this integral ∫((3x^5 −x^4 +2x^3 −12x^2 −2x+1)/((x^3 −1)^2 ))dx


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Question Number 94354 by M±th+et+s last updated on 18/May/20

$b y u s i n g o s t r o g a d s k i m e t h o d s o l v e t h i s$ $i n t e g r a l$ $\int \frac{3 x^{5} - x^{4} + 2 x^{3} - 12 x^{2} - 2 x + 1}{{(x^{3} - 1)}^{2}} d x$
Commented by MJS last updated on 18/May/20

$do you know {Ostrogradski}^{'} s Method ?$
Commented by i jagooll last updated on 18/May/20

$i want to learn this method sir$
Commented by M±th+et+s last updated on 18/May/20

$y e s s i r i r e a d e d a b o u t t h i s m e t h o d a n d i w a n t$ $t o l e a r n m o r e {i t}^{'} s a v e r y i m p o r t a n t m e t h o d$ $t o s o l v e i n t e g r a l s$
	Answered by MJS last updated on 18/May/20
	$we have p(x), q(x) are polynomes degree (p(x)) < degree (q(x)) ((p(x))/(q(x))) is a cancelled fraction ∫((p(x))/(q(x)))dx=((p_1 (x))/(q_1 (x)))+∫((p_2 (x))/(q_2 (x)))dx q_1 (x)=gcd (q(x), q′(x)); q_2 (x)=((q(x))/(q_1 (x))) now we differentiate both sides to get the constant factors ((p(x))/(q(x)))=(d/dx)[((p_1 (x))/(q_1 (x)))]+((p_2 (x))/(q_2 (x))) all we have to know is this: degree (p_i (x)) < degree (q_i (x)) in above case p(x)=3x^5 −x^4 +2x^3 −12x^2 −2x+1 q(x)=(x^3 −1)^2 q_1 (x)=q_2 (x)=x^3 −1 ((3x^5 −x^4 +2x^3 −12x^2 −2x+1)/((x^3 −1)^2 ))= =(d/dx)[((ax^2 +bx+c)/(x^3 −1))]+((dx^2 +ex+f)/(x^3 −1)) solve this for the constsnt factors a, ..., f ⇒ a=1, b=−1, c=3, d=3, e=0, f=0 ⇒ ∫((3x^5 −x^4 +2x^3 −12x^2 −2x+1)/((x^3 −1)^2 ))dx= =((x^2 −x+3)/(x^3 −1))+3∫(x^2 /(x^3 −1))dx and now it′s easy$
	$we have p (x), q (x) are polynomes$ $degree (p (x)) < degree (q (x))$ $\frac{p (x)}{q (x)} is a cancelled fraction$ $\int \frac{p (x)}{q (x)} d x = \frac{p_{1} (x)}{q_{1} (x)} + \int \frac{p_{2} (x)}{q_{2} (x)} d x$ $q_{1} (x) = \gcd (q (x), q^{'} (x)); q_{2} (x) = \frac{q (x)}{q_{1} (x)}$ $now we differentiate both sides to get the$ $constant factors$ $\frac{p (x)}{q (x)} = \frac{d}{d x} [\frac{p_{1} (x)}{q_{1} (x)}] + \frac{p_{2} (x)}{q_{2} (x)}$ $all we have to know is this :$ $degree (p_{i} (x)) < degree (q_{i} (x))$ $in above case$ $p (x) = 3 x^{5} - x^{4} + 2 x^{3} - 12 x^{2} - 2 x + 1$ $q (x) = {(x^{3} - 1)}^{2}$ $q_{1} (x) = q_{2} (x) = x^{3} - 1$ $\frac{3 x^{5} - x^{4} + 2 x^{3} - 12 x^{2} - 2 x + 1}{{(x^{3} - 1)}^{2}} =$ $= \frac{d}{d x} [\frac{{a x}^{2} + b x + c}{x^{3} - 1}] + \frac{{d x}^{2} + e x + f}{x^{3} - 1}$ $solve this for the constsnt factors a, . . ., f$ $\Rightarrow$ $a = 1, b = - 1, c = 3, d = 3, e = 0, f = 0$ $\Rightarrow$ $\int \frac{3 x^{5} - x^{4} + 2 x^{3} - 12 x^{2} - 2 x + 1}{{(x^{3} - 1)}^{2}} d x =$ $= \frac{x^{2} - x + 3}{x^{3} - 1} + 3 \int \frac{x^{2}}{x^{3} - 1} d x$ $and now {it}^{'} s easy$
	Commented by M±th+et+s last updated on 18/May/20

	$g o d b l e s s y o u s i r$
	Commented by I want to learn more last updated on 09/Jun/20

	$wow . Thanks sir$
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