Question

Integrate x/(6x-5) dx

The answer is 1/6 x + 5/36 ln abs (36x-30) + C... But I don't understand why it is 5/36 times ln.. I thought that when you factored out the 5 you would get 5 times ln abs (36x-30)..

Answer 1

I believe you are referring from your post last Thursday.

http://www.jiskha.com/display.cgi?id=1447957757

and you accept that the long division yields

x/(6x-5) = 1/6 + 5/(36x) + 25/(216x^2) + 125/(1296x^3 + ..

let's look at the term that caused your confusion

5/(36x)
= (5/36) (1/x)
integrating that give us (5/36) ln x
which was the 2nd term in my integration answer

**********

I think you should go with Steve's method which yields an exact integral of only 2 term plus a constant
(notice his 2nd term is not the same as my 2nd term, my expansion has an infinite number of terms. But both series are correct )

Just do one step of the long division to get

x/(6x-5) = 1/6 + (5/6)( 1/(6x-5)
now all you have to do is integrate those two terms

integral of 1/6 + (5/6)( 1/(6x-5) dx
= (1/6)x + (5/6)(1/6)ln(6x-5) ) + c
= x/6 + (5/36) ln ( 6x-5 ) + c

check by taking the derivative

Answer 2

and watch out for that "c"