University Calc I, Integral by substitution w/ definite integrals (natural log)

Question

It's finals time, and my pre-algebra skills are being tested while trying to complete the integral of 3(lnx)^2/x dx, from x=1 to x=11.

I have two questions:

1. Why is it that, after finding "u" to be = to lnx, we must plug in the upper limit value to the simplified u integral? to change the limits to 1 to ln(11). Hoping someone could rationalize this logic for me.
2. By the time I reach the point of substitution where I have ∫^ln(11)₁ u³, then, [u³] from 1 to ln(11), I originally found it to be (ln(ln(11)))³, but this is obviously incorrect. I have no idea how to correctly format this answer in terms of plugging in x upper limit for u.

Thanks.

Answer 1

1) You can either change the limits to be fully in the u-world or ignore the limits and integrate and then substitute u = ln(x) back in and use the original limits of 1 and 11.

By the way, the u-world limits would be ln(1) and ln(11) or 0 and ln(11) since ln(1)=0.

When doing the u=ln(x) substitution, you must also convert dx. Thus du = d(lnx)= (1/x)dx so the integral is just ∫u²du.

2)

3∫_u=0^u=ln(11)u²du = u³ |^ln(11)₀ = ln(11)³

--or--

(lnx)³ |^x=11_x=1 = (ln(11))³-(ln(1))³/3 = (ln(11))³-0 = (ln(11))³