If a=b then integral(a) = integral(b) .... 1/2lnx =/= 1/2ln(2x) ?

[Wanted]

Am I missing something?

if a = b then

Integral a = Integral b

a = dx/2x and b = dx/2x

a = (1/2) (dx/x) =
b = [dx/(2x)]

So far so good...Integral of a .. let U = x, du = dx

Integral of a = (1/2) ln|x| + C

Integral of b... let U = 2x, du = 2 dx (multiple by (1/2) to balance out numerator only being 1)
(1/2) Integral (du/u)

Integral of b = (1/2) ln|2x| + C

But wait... (1/2) ln|x| =/= (1/2) ln|2x|

So did I mess something up or is integral (a) not always = to integral (b) given a = b.

 

[fresh_42]

You scammed by hiding the difference in the constant. Your C's aren't equal.

 

[tony873004]

du = 2

du = 2 dx

 

[Wanted]

You scammed by hiding the difference in the constant. Your C's aren't equal.

Help me understand why they aren't equal. Surely there isn't a difference between using one integration technique over the other?

du = 2 dx

Yes.. updated... but unfortunately that does not change or resolve the issue here or explain any confusion.

 

[fresh_42]

Help me understand why they aren't equal. Surely there isn't a difference between using one integration technique over the other?

Sure there is. The boundaries change. If you ignore them by hiding them in the C you could add any constant value, e.g. $-\frac{1}{2}ln2$.

 

[Wanted]

Sure there is. The boundaries change. If you ignore them by hiding them in the C you could add any constant value, e.g. $-\frac{1}{2}ln2$.

Yea that's right. I just evaluated the definite integral from 1 to 2 and they were equal then. I suppose my real confusion is coming from some where else in my (parent) equation (not this one) I'll have to get back to you in a bit.