Question

Evaluate:
Integral [1,e^2] [ln(x^2) / x] dx

Answer 1

\[\int\limits_{1}^{e^2} [\ln(x^2) / x ] dx\]

Answer 2

What have you tried so far?
ln (x^2) can be written in another form, using one of the "rules of logs": ln a^b = b*ln a. Care to try that?

Answer 3

I really didn't try much to be quite honest. I didn't recognize it as any integral rule right off the bat so I just moved on for the time being. Does it have to become a u-substitution?

Answer 4

A substitution would definitely make this problem easier.
i do suggest that you rewrite ln (x^2) using the appropriate rule of logs. One of the other rules of logs is ln a + ln b = ln (a*b). Hope that triggers your memory of the third rule.

Answer 5

I will. Are you referring to lna - lnb = ln a/b?

Answer 6

If I might make a suggestion, how about \( u = \ln(x^2)\)

Answer 7

would it work if I changed the function to 2ln(x) /x , then pulled the 2 in front of the integral, and used u = ln(x)?

Answer 8

\[2\int\limits_{?}^{?} \frac{ \ln x dx }{ x }\]Yes; that's it. ln x^2 = 2 ln x. You could move that coefficient 2 outside of the integral, ending up with

Answer 9

Now try a u substitution.

Answer 10

I put u = lnx, du/dx = 1/x, du = dx/x.

Answer 11

And from there I know how to integrate it so I ended up with 4 as a final answer. Is that correct?

Answer 12

I haven't tried obtaining the definite integral, but you sound so competent that I'm willing to take your word for it.

Answer 13

Note that this is not the only way in which you could integrate. whpalmer suggested using u=ln (x^2), and upon thinking about that, I believe integration by parts would work here.

Answer 14

But the method you and I used is probably the simplest.

Answer 15

Haha well thank you, I appreciate that. And yes, I think it would work, too. This question is on a review packet from previous chapters and integration by parts I actually just learned this week so I figured it'd be best to use previous techniques.

Answer 16

It gets to be a game after a while, a fun game, to figure out which methods work best in which situations. Do let me know if you have further qu. later.

Answer 17

Fun is relative lol. I do have another one right now if you're willing.

Which of the following are antiderivatives of f(x) = \[\frac{ 3\ln^2x }{ 2x}\]?

I. \[\frac{ \ln^3x }{ 2 }\]

II. \[\frac{ 1 }{ x } lnx^2 - \ln ^2x\]

III. \[\frac{ 11 + \ln^3x }{ 2 }\]

Answer 18

If you go with my suggestion, it's \[du = \frac{2}{x} \,dx\]\[\frac{1}{2}\int u \,du = \frac{1}{4}u^2+C\] then undo the substitution and evaluate.

Answer 19

Answer choices: A) I only, B) II only, C) I and II only, D) I and III only, E) II and III only

Thanks for the suggestion, if I had more time to experiment with IBP I'd try it :)

Answer 20

Jared: Would you mind posting this new problem separately?

Answer 21

Sure

Answer 22

Just type the function to be integrated, not the possible answer choices.