Question

Which of the following are antiderivatives of f(x) = 3ln^2x /2x?

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

Answer 1

Don't bother with typing in the answer choices, unless, of course, you want to do so. I assume we could refer back to the answer choices in your previous posting.

Answer 2

I was going to in case someone else wanted to view the question.

Answer 3

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 4

I would tackle this latest integral, \[Int( \frac{ 3 }{ 2 }(\ln x)^2 dx), \]
by substitution. Why not take that 3/2 outside the integral? Then choose a substitution.

Answer 5

Forgive me for omitting the " x " that appears under the (ln x)^2 dx.

Answer 6

No problem. Okay, so I pulled the 3/2 in front of the integral and also changed the numerator to 2lnx. The 2's cancel, so we're left with \[\frac{ 3 }{ 2 } \int\limits_{?}^{?}\frac{ lnx }{ x} dx\]. Good so far?

Answer 7

Yes, except that "ln x" is squared, isn't it?

Answer 8

It was but I moved the 2 exponent in front, so that it would cancel with the 2 in the denominator.

Answer 9

Certainly I understand your train of thought, but that \[\ln ^{2}x\] is quite a different bird from \[\ln x^2\]

Answer 10

think about it for a moment. Which expression do we have in the original problem?

Answer 11

Is it really? I actually didn't know that I thought they were synonymous.
Well we have ln^2x in the original.

Answer 12

Yes. I'd vote for re-writing that as \[(\ln x)^2\]
for added clarity. Step back for a moment and then try picking an appropriate u subst.

Answer 13

Oh, I just realized what I did. I factored out the 3/2 but still wrote the denominator as 2x, therefore thinking there was a 2 to cancel.
Ok, I'll do that. One sec

Answer 14

u = lnx, du = dx/x
After integration I got ln^3x /2 so answer I.

Answer 15

that was my take also. I really like and appreciate the way you look over y our own work and find insights in doing so. glad we could discuss the differences between ln x^2 and
ln ^2 x.

Answer 16

Thanks! My main problem in math (calculus mostly) is that I go quickly and make small mistakes, so I've been trying to get better at checking things over.

Answer 17

But, is it just answer choice I? Or is there some way to get II or III too?

Answer 18

Very wise move!
At this point I think you know enough so that you could, if you wanted to, experiment to determine whether either of the other 2 answer choices are equivalent to the first one. I'd tend to say NO, but then I haven't spent much time looking over the alternative choices.

Answer 19

Yeah, it's AP Calculus so mistakes tend to be made on the rushed 45min tests haha.

I don't think III is possible, as it has an 11 in it added to our other answer, but choice II is potentially viable. I'll have to see. Thank you SO much for all of your help!