Question

I'm failing with polynomials integrals >.<'
No substitution or parts allowed.
\[\int {\frac{{3x + 1}}{{{x^2} - 4x + 3}}dx} \]

I don't need the result, I'd like someone to explain the reasoning. Thanks.

Answer 1

partial fractions

Answer 2

\[\frac{3x+1}{x^2-4x+3} = \frac{3x+1}{(x-3)(x-1)} = \frac{A}{x-3}+\frac{B}{x-1}\]

Answer 3

do u know how to find A and B?

Answer 4

Yes, I know. I actually knew how to do this method as well, and I thank you for helping out. Just one thing, my book does it in a different way, it actually wants to operate at the numerator and the denominator, I understand the steps it does, but I can't understand where it gets it from (intuition maybe?) going to write them down, sec.

Answer 5

Perhaps they are using the heaviside method

Answer 6

ok write them down:)

Answer 7

\[\int {\frac{{3x + 1}}{{{x^2} - 4x + 3}}dx} = \frac{3}{2}\int \frac{{(2x - 4) + \frac{2}{3} + 4}}{{{x^2} - 4x + 3}}dx = \frac{3}{2}\int \frac{{2x - 4}}{{{x^2} - 4x + 3}}dx + 7\int \frac{1}{{{x^2} - 4x + 3}}\]

Answer 8

there's a +7 integral of 1 over same denominator.
Then it claims
Discriminant is > 0 and does other steps, sec.

Answer 9

\[\begin{gathered}
\int {\frac{{3x + 1}}{{{x^2} - 4x + 3}}dx} = \frac{3}{2}\smallint \frac{{(2x - 4) + \frac{2}{3} + 4}}{{{x^2} - 4x + 3}}dx = \\
\frac{3}{2}\smallint \frac{{2x - 4}}{{{x^2} - 4x + 3}}dx + 7\smallint \frac{1}{{{x^2} - 4x + 3}}
\smallint \frac{1}{{{x^2} - 4x + 3}}dx = \\ \smallint \frac{1}{{{{(x - 2)}^2} - 1}}dx =
- sett - tgh(x - 2) + c \
= \\
\frac{3}{2}\log |{x^2} - 4x + 3| - 7sett,tgh(x - 2) + c
\end{gathered} \]

Answer 10

oy thats an eyesore

Answer 11

and my book result doesn't even match with wolframalpha's one... true story. More over, I have no idea what is this "sett" thingy.

Answer 12

I'll just take it as my book is wrong or something? I am kind of confused on what to do >.<'

Answer 13

no typos?

Answer 14

I'll take a scan of the page.

Answer 15

\begin{array}l
\frac{3}{2}\int \frac{{(2x - 4) + \frac{2}{3} + 4}}{{{x^2} - 4x + 3}}\\\\
\frac{3}{2}\int \frac{{(2x - 4) + \frac{14}{3}}}{{{x^2} - 4x + 3}}\\\\
\frac{3}{2}\int \frac{{\frac{14}{3}}}{{{x^2} - 4x + 3}}\\\\
\frac{3}{2}\frac{14}{3}\int \frac{{1}}{{{x^2} - 4x + 3}}\\\\
7\int \frac{{1}}{{{x^2} - 4x + 3}}\\

\end{array}

i got it to this far

Answer 16

i got no idea whay you double up the integral after that

Answer 17

here it is. by the way thank you guys

Answer 18

I see what your book is trying to do. Your book is trying to make the numerator as the derivative of the denominator plus some constant. That's why it divides the whole thing by \(\frac{3}{2}\). So they can have 2x-4, (which is the derivative of the bottom), plus some number. The way lalaly used is better anyways, esp in the case we have here.

Answer 19

tg h: tangent hyperbolic?

Answer 20

Inverse tanh.

Answer 21

What notations on earth are your book using?

Answer 22

I don't understand, is that possible that there is some book epic failure ? Because this is so weird, more over I don't get why the result is different from wolfram.

Answer 23

I have to go eat.

Answer 24

they turned it into a log for the first part to make life simpler; the last part is this


\[\int \frac{{1}}{{{x^2} - 4x + 3}}\]

complete the square

\[\int \frac{{1}}{x^2 - 4x+4-4 + 3}\]

\[\int \frac{{1}}{(x-2)^2-1}dx\]

Answer 25

now this is one of those trig integrations from way back when

Answer 26

is that french or italian?

Answer 27

That's italian.

Answer 28

wolfram does simplifications, so im sure at one point that are the same

Answer 29

It just says "Nevertheless the discriminant is > 4" and then it says "Therefore". Nothing too much important.

I see, but what's that "sett"?

Answer 30

sett is 7

Answer 31

-7 tg h yada yada

Answer 32

maybe lol ... but i see that its not ....lol

Answer 33

so it's kind of a printing error or something? Why in the hell they'd write "sett"? More over in italian it's "sette" and ... I don't know :D . LoL.

Answer 34

http://www.wolframalpha.com/input/pod.jsp?id=MSP18419i891ahbaffcbce000043d5e35f5ed1d367&s=9&button=1

oh there it is the tanh!

Answer 35

http://www.wolframalpha.com/input/?i=integrate%20%283x%2B1%29%2F%28x^2-4x%2B3%29&t=mfftb01

Sowwy, show steps! ^

Answer 36

yep

tanh^(-1) (2-x)

and the book took out the negative and stuck it in front

Answer 37

Ohhhh... I just got what "sett" means. Sector tangent x. Which is the inverse function of the iperbolic tangent

Answer 38

Got it, all righty, some weird steps it does by the way, I'd have used lalaly method and I guess that'd have been simpler.
Thank you all guys, that's really appreciated.

Answer 39

Goodluck:D

Answer 40

I don't think it's weird, this method can be used in situations where finding constants of the partial fractions is difficult. I came across such a problem last week and used the same approach the book used, (i.e make the top the derivative of the bottom so you get a log for the integration).

Answer 41

And good luck! :)