Dr. Brown Assigned:
7.1 8,10,26,32,42,48,57,68
7.2 2,10,20,26,38,42,56,68
Tuesday, February 7, 2012
Class Discussion due 20120214
Check out the discussion that's already happened here: Class Discussion Due 20120207
If you didn't already participate in that discussion, please do. Participating is easier than you think.
A new thing you can comment on is what you like or don't like about section.
Last week I had you look at Appendix A and Reference page 1 Algebra.
This week, look at the left-hand side of Reference page 2.
Try to finish reading Appendix A and do some problems (every 8 is a good suggestion).
If you tried to post, but couldn't feel free to send your comment by e-mail. Hopefully you'll eventually be able to post on your own. Sometimes it's a matter of web browser compatibility. Personally I use Google Chrome when I browse on www.blogger.com.
If you didn't already participate in that discussion, please do. Participating is easier than you think.
A new thing you can comment on is what you like or don't like about section.
Last week I had you look at Appendix A and Reference page 1 Algebra.
This week, look at the left-hand side of Reference page 2.
Try to finish reading Appendix A and do some problems (every 8 is a good suggestion).
If you tried to post, but couldn't feel free to send your comment by e-mail. Hopefully you'll eventually be able to post on your own. Sometimes it's a matter of web browser compatibility. Personally I use Google Chrome when I browse on www.blogger.com.
Thursday, February 2, 2012
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.
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 $\frac {d}{dt} \cos (t)=-\sin(t)$.
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.
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.
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