Question

Is there a way to access the function writing tool when first posting a question or only when you answer? But more importantly:

Integrate: x/(x-6) using partial fractions and the answer book says the first step is :
Integrate: ((x-6) + 6)/(x-6).....wha?

Answer 1

that would give you 1 + 6/x-6...which is easier..the book is right

Answer 2

Yes, because
\[ \frac{x}{x-6} = \frac{x-6}{x-6} + \frac{6}{x-6} = 1 + \frac{6}{x-6} \]

Answer 3

the integral of that is 6ln(x-6) just so you know :)

Answer 4

+C...forgot the arbitrary constant

Answer 5

you can effectively do the same thing if you had just used substitution:
u = x-6
du = dx

--> integral (u+6)/u = 1 + 6/u = 1+ 6/x-6

Answer 6

Sorry kind of slow but for some reason my mouse went on strike??!1 <reboot>
but why should I even think to add what effectively amounts to zero to the numerator? I see after doing that, that it might be easier. But in the whole scheme of things there could be an infinite number of things I could do....how would I know that this action would make it easier? Is this like a math pun to math people?

thanks for help.

and then this is just me whinging: why would you make that part of the first freaking question in the exercises on how to do partial fractions? grrrrr

Answer 7

Like just about everything in life, you have to learn techniques. This is just a technique for dealing with this sort of problem.