E vs. Coco (If you don't know math, just skip the mumbo-jumbo. You'll get the story regardless.)

In my school, there's a tutoring center, where you can get help with math and physics. They lady in charge has this nasty high-pitched voice, painted-on eye lashes, and wears the horriblest, loudest, most-mismatched clothes you've ever seen. When you come in she screams, "Sign in! Sign in!" Imagine what would happen if people didn't sign in at the tutoring center... or even worse, they might sign in but leave out the last four digits of their social security numbers... Oh! God save us from such horrors! Thank God we have Coco to prevent such tragedies.

Despite the b*tch at the door, I like studying there. The atmosphere is kind of like a zal, and it's harder to space out or get too involved with checking my email on my phone when I'm with other people.

Anyhow, Coco had just finished screaming at one of the tutors for not offering his help. (That exchange was actually kind of funny.
Coco: Why you don't ask students if they need help?
Tutor: I aksed 'em all, two minutes ago.
Girl on the other side of room: (raising her hand) I need help)

After that exchange, Coco felt like making herself useful before she resumed her post at her computer to look out for students trying to sneak in without signing in. So she moseyed over to me and asked if needed help. I accepted her offer, and we started working through a problem together. My trig is horrible. I learned it all myself, so I'm missing lots of basics.

Me: (hoping she'll tell me the formula) OK, now it's cos2t = 0, so we need the double angle formula...
Coco: (mutters something unintelligible with her foreign accent)
Me: Yeah, the double angle formula, which is...
Coco: (smirking) the double angle formula
Me: which is...
Coco: (smirks silently)
I turn to cheat sheet on the inside from cover, and shamefacedly copy the double angle formula. But I copied the wrong one. So I copy the second one, even more shamefacedly.

I wasn't going to let this slide. I figured I'd ask her a question that's been bothering me since yesterday.

∫ e^[ln(2x)] dx Looks like a scary integral. It's not.
∫ e^[ln(2x)] dx = ∫ [e^ln(x)]^(2) dx
= ∫ x^2 dx
= x^3/3 + c

But suppose you didn't simply it first, could you still solve the problem by taking the ln of both sides? Here's what I did yesterday:
y = ∫ e^[ln(2x)] dx
ln(y) = ln ∫ e^[ln(2x)] dx
= ∫ ln e^[ln(2x)] dx
= ∫ [ln(2x)] dx
u=2x; du=2
= 1/2 ∫ [ln(2x)] 2dx
= 1/2 ∫ ln(u) du
using integration by parts, you get
1/2 [x*ln(2x) - x] + c

remember, that all this is ln(y). To get y, we raise that whole mess to e, so the final answer is:

e^(1/2 [x*ln(2x) - x] + c)

This is obviously not the right answer. But what had I done wrong? Perhaps the mistake was taking the ln of both sides?

I showed Coco the original expression.
Coco: That simplifies to X^2.
Me: Yeah, but if I don't simplify it, could I get the integral by taking the ln of both sides, and then raising everything to e?
Coco: Don't do it.
Me: I understand it's harder, but is it allowed?
Coco: Don't do it.
Me: Yeah, but could I? Suppose you have an integral that you can't evaluate, are you allowed to take the ln and then raise it to e?
Coco: Yes, but-
Me: Great. So I want to see how you would do that here.
Coco: No
Me: But I want to see how it would work out. Could you show me?
Coco: No

There I was, thirsting for knowledge, and she was refusing and making me feel like a spoiled child who wants the parent to give it unnecessary stuff.

So I bided my time and thought evil thoughts about Coco. And thought about this rule some more, and realized that Coco was wrong. Here's a simple proof:
∫2^x dx
According to the Coco Theorem, you proceed as follows:
∫2^x dx = y
∫ln 2^x dx = ln(y)
= ∫ x*ln2 dx
ln2 is a constant, so you take it outside:
= ln2 ∫ x dx
= ln2*x2/2 + c. That is complete BS. The Coco theorem is wrong

But by now Coco was ensconced near her computer, heavily engrossed in Yahoo news. I showed her my work and asked her what I had done wrong? Each step had been legal, according to her, but the answer was wrong. Everyone knows that ∫2^x dx = 2^x/ln(2) + c. Where had we gone wrong? She got all confused. She started to scribble, pulled a bunch of u-substations but couldn't get anywhere. She asked for my book, and started to flip through chapter 7. I calmly reminded her that Logarithms and exponentials are discussed in chapter 6. She violently flipped to chapter six, where they were discussing the derivatives of exponentials. I knew that the formula for integrating exponentials was on the next page, but I kept quiet. Let the master of the tutoring center plumb the mysteries of mathematics in peace.

She scribbled some more, and decided that ∫2^x dx is in fact 2^x/ln(2) + c, as I had predicted. So what was wrong with what I had done? Is it possible that you're not allowed to take the ln of both sides and the raise the equation to e? Eh? Was I perhaps misinformed a few minutes ago? Oh yeah, I had suspected as such. Well, Coco, don't feel bad. I hear you got the double-angle formula down pat.