Integral Simplifying

For the following question have I used the correct method to simplify the integral? - I just took 1/a out as a constant and cancelled stuff.

but I don't know if that's the right approach as they use d(az) which means a isn't constant?
But I still ended up with the same answer so have I done something wrong?

original question, part a: https://isaacphysics.org/questions/reciprocal_integration?board=5905a7a8-3e53-4464-bbc0-e8fb650c8115&stage=a_level

Reply 1

mosaurlodon

Working:

Reply 2

mqb2766

Original post

by mosaurlodon

Working:

Their working is fine as you can integrate wrt any varible so x, 3x, .... but you have to do the transformation (3 parts, integrand, integration variable, limits) correctly

Your working isnt quite correct as doing a sub z=x/a
a dz = dx
with z limts from 1 to b so
Int_1^b 1/z dz
Youve used the original x limits and forgot about the dx = a dz part and the 1/a doesnt cancel with the a in the limits like youve done. Luckily the mistakes cancelled.