Need help with 2 integrals.

[quackzilla]

Hey, I have a crappy community college professor and apparently I am not smart enough to figure out the parts she didn't teach.

the definite inegral of (ln x)^2 - 1 (part of a geometry problem)
from e^-1 to e

I tried this:

[(x ln x - x)^3]/3 - x

And got 2.48, but looking at the graph it looks like it should be about 1.5.

This is just a homework problem, but none of the class can get it.

I also have another question on a different problem.

a definite integral from -.5 to .5

SQRT[1-(2 cos (pi*x))^2]

I got as far as SQRT[1 - 4pi^2sin^2(pi*x) and I have NO idea how to integrate that.

None of the class could figure out this one either.

My calculus teacher is about 75 years old, and she shouldn't be teaching, I feel like I haven't learned anything except what I have taught myself.

Thank you for any help!

 

[mattmns]

Could you post the original problems?

Also, I am not sure what you mean by:

I tried this:

[(x ln x - x)^3]/3 - x

And got 2.48, but looking at the graph it looks like it should be about 1.5.

If you evaluate the integral you should get:

$$\int_{e^{-1}}^{e}\left( (\ln x)^2 - 1 \right) dx \approx -1.47$$

Is that what you mean by 1.5? Do you need help evaluating the integral? What are you having trouble with?

 

[quackzilla]

yeah, it looked by eye like it should about 1.5, but the equation that I got when I integrated, what I posted above, gave me about 2.5.

I'm just not sure how to integrate the square of (log x)...

 

[mattmns]

If you can, use a table (there should be one in the back of your book). If you want to, or have to, do it by hand, try substitution and then use integration by parts (twice ). There may be a more clever way to do it, but what I said works (and it is not that bad).

 

[Gib Z]

$$\int\left( (\ln x)^2 - 1 \right) dx = x(\ln x -1)^2 + C$$

That makes your definite integral $$- 4(e^{-1})$$, which is approx
$$-1.47151776468576928638209508064584346978324452 412707133803134720678984598297959921342858909738367 857498650930110737598083298790317116051603450661435 797639513236877746950935246019455596450058246537995 087991473790383175898921019956998181295746483185924 205859008244917689235665970626640146029830913482213 141495355242721915044782731957381798940295727439686 646973201427974883282840911007208633997693512676286 267068704934643292150449166249502883788002816203893 3702710305010112121507544660628374615198170962548287151378$$

 

[quackzilla]

thanks for the help!

 

[dextercioby]

Do you know how to part integrate ?If so, then your first integral shouldn't be difficult.

$$\int_{1/e}^{e} \ln^{2} x \ {}dx -\int_{1/e}^{e}{}dx$$

=to be solved -(e-1/e).

 

[dextercioby]

Also, are you absolutely sure about the form of the second integral ? Cause I'm getting a nasty combination of elliptic integrals.