I have done Int(1/ax)dx=1/a*Int(1/x)dx
=[(log x)/a] + C.
The working in the book is
Int(1/ax)dx=1/a*Int(a/ax)dx
=[(log ax)/a]+C.
Which one is correct and where is the error in my working, if so?
both are correct
log ax = log a + log x
so, the loga/a is included in the C when you do it. The book probably did it their way to retain the appearance of (ax) as a function of x.
int(1/ax) dx let u = ax and you have
1/a int 1/u du
integrate that to get 1/a logu + C
= 1/a log(ax) + C
The C's are different, that's all.
FWIW, I prefer your way.
Thank yoy very much Mr. Steve for making it so clear and easy.
To determine which solution is correct and identify any errors, let's analyze the steps you took and compare them to the steps in the book's working.
Your working:
1. Integrate 1/ax wrt x: ∫(1/ax)dx = (1/a) * ∫(1/x)dx
2. Simplify: ∫(1/ax)dx = (1/a) * ln|x| + C
The book's working:
1. Rewrite the integrand as a/ax: ∫(1/ax)dx = (1/a) * ∫(a/ax)dx
2. Integrate (a/ax) wrt x: ∫(a/ax)dx = (1/a) * ln|ax| + C
(Note: Here, the book uses the property ln(ab) = ln(a) + ln(b))
Comparing the two solutions, we can see that the book's solution is indeed correct. The error in your working lies in the simplification step, where you used the natural logarithm (ln|x|) instead of the logarithm of the absolute value of the argument of the logarithm (ln|ax|).
When integrating "1/x" with respect to x, the result is ln|x| since the natural logarithm of the absolute value of x is used to account for the potential sign changes as you move across positive and negative x-values.
However, when integrating "(a/ax)" with respect to x, the argument inside the logarithm is "ax," not just "x." Thus, the correct result is ln|ax|.
To summarize, the correct solution is [(ln|ax|) / a] + C according to the book's working. The error in your working was using ln|x| instead of ln|ax| in the simplification step.