anonymous
  • anonymous
James?
Mathematics
chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
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anonymous
  • anonymous
give me edals
JamesJ
  • JamesJ
so the answer is?
anonymous
  • anonymous
oh that was annoying i cldnt type

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

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