Integral of x*exp(x)*cos(x) with respect to x

This post is a follow-up to Useful Mnemonic: DETAIL where the problem was posed.

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Determine $f(x)=\int x e^x \cos (x)dx$.

CHOICE A (Suggested by DETAIL):
Let $u=x\cos(x)$ and $dv= e^x dx$.
Then $du=\cos(x)-x\sin(x)dx$ and $v=e^x$.
Then $f(x)=x\cos(x)e^x - \int e^x \left[ \cos(x)-x\sin(x) \right] dx$

CHOICE B:
Let $u=x e^x$ and $dv=\cos(x)dx$.
Then $du = e^x+x e^x dx$ and $v=\sin(x)$.
Then $f(x)=x e^x \sin(x) - \int \sin(x) \left[ e^x +x e^x\right] dx$

CHOICE C:
Let $u=e^x$ and $dv=x \cos(x)dx$.
Then $du = e^x dx$ and $v=\int x\cos(x)dx$
Let $\hat{u}=x$ and $d\hat{v}=\cos(x)dx$.
Then $\hat{v}=\sin(x)$
Then $\int x\cos(x)dx = x \sin(x)- \int\sin(x)dx = x\sin (x)+\cos(x) +\hat{C}$
We only concern ourself with a particular $v$, so we assume $\hat{C}=0$.
Then $f(x) = e^x \left[ x\sin(x) + \cos (x) \right] -\int \left[x\sin(x)+\cos(x) \right] e^x dx$

What remarks can we make? For this specific problem, regardless of the above three choices for $u$ and $dv$, we will have to do integration by parts again. DETAIL recommended choice A, but choice B was just as short. Though as a personal opinion, I find it easier to make sign errors when determining antiderivatives to trigonometric functions than when determining antiderivatives to exponential functions.

Recall that $\frac{d}{dt} \sin (t) = \cos (t)$, while $\frac {d}{dt} \cos (t)=-\sin(t)$.

Useful Mnemonic: DETAIL

[20120202]
A student had an interesting random thought.

What to do when the integrand is a product of three functions?

In particular he asked about $\int x e^x \cos (x)dx$.

What do you think?

[20120207]
I had you work on this in class. Please complete the problem and ask questions if you have any.

Class Discussion due 20120207

#2. In the comment section of the relevant week, submit a question or something you learned every week. These should be mildly relevant to the class. If you posted something along these lines on the Facebook page, you may simply copy and paste what you posted. Other comments such as interesting links, even those not related to math are welcome, but do not count toward the required participation. No duplicates, so the earlier you submit a question or comment, the less you'll have to read of other posts.

Read Appendix A.
Memorize/Learn Reference Page 1: Algebra.

[20120202:] If you don't want to sign with your name, that's fine. Next week in class, I will figure out which user names belong to which people.

Math 109 Spring 2012

#0. My name, email, class website, other website. I drew a map of where my office room and math help room are located.
#1. Staple your homework or else lose a point.
#2. In the comment section of the relevant week, submit a question or something you learned every week. These should be mildly relevant to the class. If you posted something along these lines on the Facebook page, you may simply copy and paste what you posted. Other comments such as interesting links, even those not related to math are welcome, but do not count toward the required participation. No duplicates, so the earlier you submit a question or comment, the less you'll have to read of other posts.
#3. I'd like to try to get you to read sections of the appendix and do extra problems. Sometimes it's some of these basic ideas that students trip up on, creating an obstacle to the ideas the instructor wishes to teach about calculus. Related is reference page 1, 2, 5, 6, 3, and 4.
#4. We will try working in groups and form strong class participation. I'd like to help you learn, so it's up to you to ask questions that help me figure out how to run the section. I prefer a more dynamic feel to class.
#5. I put a strong emphasis on you knowing your trigonometric identities.
#6. Some useful sites: Wikipedia, Mathworld, Wolfram Alpha. But I do explain that we shouldn't let calculators and Wolfram Alpha be crutches. I cry on the inside when I see a student punch in something like 20 divided by 5 into a calculator.
#7. I explain my goal of applying math to more than just it's uses as math, but as about thinking. Like back in the day with geometry proofs.
#8. A pattern... or is it?
1,2,1,2, (1 is most popular, some 3's)
1,2,1,2,3,1,2,3, (4 is most popular, but some 5's)
1,2,1,2,3,1,2,3,5,1,2,3,5,? (7 and 8 are most popular. Good reasons for other answers.)
The point is about expectation of a pattern where there isn't.
[1]#9. pg A10 #70.
$-\pi+\pi=0$
$\sqrt{2}\cdot \sqrt{2}=2$
$\pi\cdot\frac{1}{\pi}=1$
[2]#10. Help Room 10 minutes.