Well first 2 week of school done!
One thing I'm comfortable with is L'Hopital's Rule. It is used when an indeterminate form occurs with a limit. It could be in any form of: infinity-infinity, 0/0, infinity/infinity, 0(infinity), 0^0, 1^infinity, and infinity^0.
When you get an indeterminate form, you take the derivative top then derivative of bottom. If you still get an indeterminate form just repeat the second step.
EXAMPLES:
lim e^2x - 1/x = e^2(0) - 1/1 = 1-1/0 = o/o <---indeterminate form
x-->0
So, take derivative of top then bottom.
e^2x - 1/x = 2e^2x/1 = 2(1)/1 = 2
lim lnx/x = infinity/infinity <---indeterminate form
x-->infinity
1/x/1 = 1/x = 0
With this example you would use your limit rules because it is as x approaches infinity. Since the degree of bottom is larger than the degree of top it equal zero.
Another thing we have covered is substitution. I can usually pick out my u and du, but sometimes I have trouble completing the problem. I guess I just need more practice.
EXAMPLE:
S tsint^2dt
u=t^2 du=2t
1/2 S sinudu
-1/2cost^2 +C
By parts is also something I can usually get my u, du, dv, and v. I just sometimes have a hard time finishing the problem. The formula is: uv - Svdu. Remember that the u is something that you want to reduce and dv can be the dx.
EXAMPLE:
S xe^x
u=x v=e^x
du=1 dv=e^xdx
xe^x- S e^xdx
xe^x-e^x +C
What I need help on! The whole sin, cos, secant, tan stuff with the formulas. I really didn't get the homework...
How would you work this?
S sec^4 5xdx
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okay. So you see from the tan colored box that this matches none of those right? but think of this:
ReplyDeleteS sec^4(5x) is the same as:
S [sec^2 (5x)][sec^2 (5x)]
because the squares add togive you the exponent of 4 right? okay.
(tidbit: anytime I see an exponent greater than 2, I automatically pull out a squared whatever...that might help)
So, from our formula sheets, you see that sec^2=1+tan^2. This means that I can put this identity in for one of the sec^2. So overall this gives you:
S (1+tan^2(5x))(sec^2 (5x))
Now, what I do is foil out everything and integrate each thing separately:
S sec^2(5x) + S tan^2(5x)sec^2(5x)
So for the first part, you know that the integral (after having your u=5x and du=5) equals tan(5x).
For the second part you see that sec^2 is the derivative of tan...so your u=tan(5x) du=5sec^2(5x).
Now after accounting for all the du's and everything you should end up with:
1/5tan(5x) + 1/15tan^3(5x). Got it? okay byes!!!