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JamesJ
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so the answer is?
rld613
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oh that was annoying i cldnt type
neverforgetvivistee
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give me medals free medals for everyone
rld613
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can you write the equation again please :D
JamesJ
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\[ \frac{d \ }{dx} \int_{17}^x \sin(t^2)/t \ dt \]
rld613
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d/dx(sinx^2/x) - d/dx(sin(17)/17)
JamesJ
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No
rld613
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where did i go wrong?
JamesJ
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look at the earlier example with ln
rld613
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k i will
rld613
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wasn't it d/dx(F(x)-F(17))
rld613
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oh i see where i went wrong!!!!
JamesJ
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Yes, but F is not the integrand.
rld613
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F(x)=-cosx/x???
JamesJ
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Nooo
rld613
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K just tell me the answer
JamesJ
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4
Look at the ln example again.
rld613
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In the example we subsituted t for x
rld613
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In the example we subsituted t for x
JamesJ
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4
\[ \frac{d \ }{dx} \int_2^x \ln(t^2+1) \ dt \]
JamesJ
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What is the value of that expression?
rld613
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i dont know i am feeling very stupid!!!!!!
dF/dx=ln(x^2+1)
JamesJ
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If \[ F(t) = \int \ln(t^2+1) \ dt \] then
\[ \int_2^x \ln(t^2+1) \ dt = F(x) - F(2) \].
Hence
\[\frac{d \ }{dx} \int_2^x \ln(t^2+1) \ dt = \frac{d \ }{dx} (F(x) - F(2)) \]
rld613
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righ
JamesJ
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and by the Fund Theorem of Calculus, dF/dx = ln(x^2 + 1).
Also (d/dx)F(2) = 0 because F(2) is a number, a constant.
Hence the entire expression is just ln(x^2+1)
missyfredtom
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James do you think you help me please?
JamesJ
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4
So, return now to the next example where the integrand has a sin function. What is the value of that derivative?
JamesJ
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@mft, I'll try and get to it. unfortunately you're not the first to ask.
rld613
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dont take james away form me:(
missyfredtom
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i can wait
JamesJ
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quick rld613. I want to move on with you to yet another example afterwards
Hero
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rld said she didn't need OS anymore
JamesJ
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so (d/dx) int_17^x sin(t^2)/t dt = ...
rld613
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i didnt get that why is 17 to th epower of x
JamesJ
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\[ (d/dx) \int_{17}^x \sin(t^2)/t \ dt \]
rld613
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yes
JamesJ
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evaluate. What is the that equal to.
rld613
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y is it wrong what i showed u b4?
rld613
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plz be nice to me and show me all the step? :D
JamesJ
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No. You write out the steps, following line by the line the example we just used for ln. I'm going to help someone else for a while and expect the right answer when I get back ;-)
rld613
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LOL see u later :D
JamesJ
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I'm serious. And I'll be back in 5 minutes, so hurry.
JamesJ
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4
??
rld613
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u did u come back already!!!
JamesJ
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4
What's the answer?
rld613
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K can u walk me thru the steps cuz like i bever learnt this stuff?
YOu r so scary LOL
rld613
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I was hoping i wld have the answer b4 u wld return
rld613
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teh answer wld sinx/x?
JamesJ
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Exactly as before, exactly, let
\[ F(t) = \int \sin(t^2)/t \ dt \]
Then \[ \int_{17}^x \sin(t^2)/t \ dt = F(x) - F(17) \]
Thus the derivative of this definite integral
\[ \frac{d\ }{dx} (F(x) - F(17)) = ... \]
rld613
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iSo what is F(x)?????
rld613
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That is where i am going wrong
rld613
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oh i know si(x)?
rld613
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d/dx(si(x)?
JamesJ
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like we've seen, we actually don't need an explicit formula for F(x). We just need to know it is this integral and then use the Fundamental Theorem of Calculus.
JamesJ
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What is dF/dx here? What is it equal to?
JamesJ
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By the FTC, dF/dx = .... what?
rld613
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You are forsure ready to kill me!!!!!!!!!!!!!!!
rld613
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You are forsure ready to kill me!!!!!!!!!!!!!!!
Hero
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shakes head
rld613
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dF/dx= sinx^2/x
JamesJ
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4
correct. Hence the derivative wrt (with respect to) x of the entire integral equals what?
pokemon23
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hey james can you help me with math?
JamesJ
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This thing ... what's it equal to?
rld613
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sinx^2/x????? this is a guess
JamesJ
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it shouldn't be guess. it is exactly right.
rld613
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oh ya !!!!!!!!
rld613
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That is what i thought in the first place but i left out by accident the squared
rld613
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I can be so annoying sometimes
JamesJ
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4
Now one more to make sure you really understand.
Evaluate this:
\[ \frac{d \ }{dx} \int_x^{x^2} \sqrt{t^3+1} \ dt \]
rld613
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oh u r so funny!!!
JamesJ
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Now write out the steps. This problem isn't exactly the same as the others.
rld613
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ok give me a sec
rld613
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\[d/dx=(\sqrt{x ^{6}+1}-\sqrt{x ^{3}+1})\]
rld613
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This one i guessed
rld613
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I think it was suppossed to be dF/dx
JamesJ
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The expression is equal to
\[ \frac{d \ }{dx} \left( F(x^2) - F(x) \right) \ \ \ \ \ \ --(*)\]
where by the Fund Theorem of Calculus, \[ dF/dx = \sqrt{x^3+1} \]
Now given that, evaluate the first expression, (*).
JamesJ
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hint: you need to use the chain rule.
rld613
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ok so let me try that
rld613
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james u gotta help me out on this one
JamesJ
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you think about it for a bit and write the answer sometime in the next day when you know what's going on.
rld613
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alrighty thanks james
rld613
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I will call you back when i get the
rld613
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I got the answer :D
rld613
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\[2x \sqrt{x ^{6}+1} - \sqrt{x ^{3}+1}\]
Here is my answer.
Thanks james for your help. I really appreciate it
JamesJ
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that's it. Good
JamesJ
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One more variation they might throw at you in the exam. Evaluate:
\[ \frac{d^2 \ }{dx^2} \int_2^x \ln(\cos t) \ dt \]
rld613
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Oh this is very simple there is just one small catch that you have to differentiate it twice
rld613
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d/dx(ln(cosx))
(1/cosx)*-sin(x)= -sin(x)/cos(x)
-tan(x)
JamesJ
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exactly, good.
rld613
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I think that is the answer. I took my final exam already and it was basically simple except ofr a related rates problem. Going on vacation for a month So see u then. Thanks for your help james
rld613
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oh u are here LOL
rld613
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Thanks for helping me out
JamesJ
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sure. have a good vacation.
rld613
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U better take a vacation from OS tooo!!! :D
JamesJ
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I might just do that.
rld613
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Ya u need a break to refresh again
rld613
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OS wldnt survive without James myinaya satellite and Hero