Please check my work on this integral

Hi, please check my work. Here is the problem:

(integral sign evaluated from 6 to 3) dy/(y-4)^2

let w = y - 4
dw = dy

change limits in terms of w:
when y = 6: w = 6 - 4 = 2.
when y = 3: w = 3 - 4 = -1.

(integral evaluated from 2 to -1) 1/w dw

[ln|w|] evaluated from 2 to -1

= ln|2| - ln|-1|

= ln|2| - 0

Final answer = ln|2|.

Thank you!

That would be right, but you've missed a crucial part of this integral, it's improper. The integrand doesn't exist at y = 4, so you have to split the integral there and do it twice. Fortunately, everything besides the evaluation is correct.
∫[3,6] dy / (y - 4)²
∫[3,4] dy / (y - 4)² + ∫[4,6] dy / (y - 4)²
lim[b->4⁻] ∫[3,b] dy / (y - 4)² + lim[a->4⁺] dy / (y - 4)²
[insert all your work here]
w(b) = b - 4
w(a) = a - 4

††[Edit: Ignore all the integrands, replace with dw/w² and re-perform the integration. Note: It will still diverge.]††
lim[b->4⁻] ∫[-1,b-4] dw / w + lim[a->4⁺] ∫[a-4,2] dw / w
lim[b->4⁻] (ln|b - 4| - ln|-1|) + lim[a->4⁺] (ln|2| - ln|a - 4|)

At this point you notice that both limits result in an infinite term. This means that the integral diverges.

When either one of the limits are infinite, or the integrand doesn't exist somewhere inside or on the boundary of the interval of integration, the integral is known as "improper" and you have to use this limit method to deal with it.

Edit: Wow, I missed a gigantic detail there, haha. So just replace all the w's with w²s and redo the integrals. All the improper, limit stuff is correct though.

You forgot that it's (y - 4)²
so the substitution should be:
(integral evaluated from 2 to -1) 1/w² dw

-1/w evaluated from 2 to -1
(-1/2) - (-1/-1) = -1/2 - 1 = -3/2

I believe you've got it now.

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