Question

can someone show me how to use long division to rewrite (x^2-3x+2)/(x+1) and is there any way of knowing if you need to use long division on an integral?

Answer 1

Attempting long division on integrals, more likely than not, will make it harder :)

Answer 2

so you see that x^2 -3x +2 can being factorized easy like (x-2)(x-1)
- and now because there not is like factor (x+1) so hence this signing that you need using long division

Answer 3

thx johnny and terenz long division is sometimes the best option

Answer 4

but if the denominator is just a linear function of the form
ax + b,
and the numerator is another polynomial, then
I'd just use integration by substitution, that way, no tedious long division (I'm not good at it :) )

Answer 5

yeah but that doesnt always work like take the integral of the function i mentioned and youll see that u substitution and factoring dont get you anywhere

Answer 6

I'll try :)

Answer 7

\[\int\limits_{}^{}\frac{x^{2}-3x+2}{x+1}dx\]

Answer 8

I let
u = x + 1
du = dx
\[\int\limits_{}^{}\frac{(u-1)^{2}-3(u-1)+2}{u}du\]
\[\int\limits_{}^{}\frac{u^{2}-2u+1-3u+3+2}{u}du = \int\limits_{}^{}\frac{u^{2}-5u+6}{u}du\]
\[\int\limits_{}^{}(u - 5 + \frac{6}{u})du= \frac{u^{2}}{2}-5u+6\ln|u| + C\]
\[\frac{(x+1)^{2}}{2}-5(x + 1)+6\ln|x+1| + C\]

How's that? :)

Answer 9

that is a very complicated u substitution and in the time it took u to do it i taught myself long division and solved the integral

Answer 10

To be fair, what took a long time was typing it up

Answer 11

still though this would be the most complicated u sub ive done so far and this long division isnt as hard as i thought it would be before

Answer 12

It doesn't matter, as long as the process is legit and you arrive at the same correct answer-
and besides, you said I couldn't do it by u substitution ;)

Answer 13

and the answer in the book and my answer came out to (x^2)/2 -4x +6ln(x+1)+C which is probably the same answer as yours just more simplified

Answer 14

and i didnt forget the absolute bars i just cant do em on here

Answer 15

It's the same, lol, and yeah, more simplified, I didn't feel like simplifying further :D

Answer 16

soo i arrived at my answer faster and more simplified which is crucial on tests so now i have another method in my arsenal to tackle integrals!

Answer 17

I'd simplify it if it was on paper, though.
And uhh, you being faster is debatable... :)

Answer 18

well if i did it your way it would have taken ME longer but seeing it done your way will help me if i forget how to do long division so thx :)

Answer 19

uhh, sure no prob... (btw, I never imagined using long division for integrals, just a pet peeve, I guess)

Answer 20

i never imagined using long division past elementary school :P