Question

Finding general term for :

\[ \int \cfrac{px+q}{ax^2 + bx + c} dx \]

Answer 1

@iambatman :-)

Answer 2

My notifications are off so I just realized you tagged lol

Answer 3

:-) Make the notifications on for me lol

So, let us start this problem.

Answer 4

Bleh it's a nasty one, but I guess you want to rewrite the integrand.

Answer 5

I personally will like the coefficient of x in numerator and x^2 in denominator to become one.

That is : \[\int \cfrac{ x + \cfrac{q}{p} }{x^2 + \cfrac{b}{a} x + \cfrac{c}{a} } \times \cfrac{a}{p} dx \]

Answer 6

@ganeshie8 will also be able to help us! :-)

Answer 7

Sorry, will it be p/a ? (just messed up everything :P )

Answer 8

\[\int\limits \frac{ p(2ax+b) }{ 2a(ax^2+bx+c) }-\frac{ bp-2aq }{ 2a(ax^2+bx+c) }dx\]

Expressed as a sum of fractions, you get that...

Answer 9

\(\\ \text{9. To integrate }\quad \huge \frac{ax+b}{\sqrt{px^2+qx+r}} or \frac{ax+b}{px^2+qx+r} \\ \large express \quad ax+b=A\frac{d}{dx}(px^2+qx+r)+B \\ \text{then separate D.}

\\~ \\~ \\~ \\~\)

D = Denominator

Answer 10

How did you get that? @iambatman

I'm not sure that how did you do that.

Answer 11

Oh.. hartnn got something from his tutorial for us.

Answer 12

I think I may have done it wrong, just ignore it lol, just use what hartnn said., it's much easier.

Answer 13

Yeah, what hartnn suggested was there in my book, but it was not well explained. I really don't know how they get these substitution ideas.

Answer 14

I think I got how hartnn got that.

Answer 15

you have to know the formulas.
we have this formula :
\(\\ \huge \text{In general,}\int \frac{f’(x)}{f(x)}\:dx=\ln|f(x)|+c
\)

which means if denominator is quadratic,
we need to express numerator in terms of derivative of denominator to be able to use this formula

Answer 16

also we have formulas for
1/(x^2+a^2) , 1/(a^2-x^2) and so on

so even if we have a constant in numerator and quadratic in denominator, we should be able to solve it

Answer 17

@mathslover what does it show on how they achieved the formula?

Answer 18

I know of it, but I never knew how to derive it.

Answer 19

we always try to express the given problem in terms of standard formulas.
to be able to do that, we need to know all standard formulas which we can use.
to know those, we need to practice good amount of problems on every known.

Answer 20

on every known formula*

Answer 21

Hartnn, yes, that was a great explanation even.

Answer 22

I am getting it now... !

Answer 23

Got it, thank you both :)!

Answer 24

:-) Thank you @iambatman :)