- anonymous

James?

- chestercat

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- anonymous

give me edals

- JamesJ

so the answer is?

- anonymous

oh that was annoying i cldnt type

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## More answers

- anonymous

give me medals free medals for everyone

- anonymous

can you write the equation again please :D

- JamesJ

\[ \frac{d \ }{dx} \int_{17}^x \sin(t^2)/t \ dt \]

- anonymous

d/dx(sinx^2/x) - d/dx(sin(17)/17)

- JamesJ

No

- anonymous

where did i go wrong?

- JamesJ

look at the earlier example with ln

- anonymous

k i will

- anonymous

wasn't it d/dx(F(x)-F(17))

- anonymous

oh i see where i went wrong!!!!

- JamesJ

Yes, but F is not the integrand.

- anonymous

F(x)=-cosx/x???

- JamesJ

Nooo

- anonymous

K just tell me the answer

- JamesJ

Look at the ln example again.

- anonymous

In the example we subsituted t for x

- anonymous

In the example we subsituted t for x

- JamesJ

\[ \frac{d \ }{dx} \int_2^x \ln(t^2+1) \ dt \]

- JamesJ

What is the value of that expression?

- anonymous

i dont know i am feeling very stupid!!!!!!
dF/dx=ln(x^2+1)

- 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

righ

- 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

James do you think you help me please?

- JamesJ

So, return now to the next example where the integrand has a sin function. What is the value of that derivative?

- JamesJ

@mft, I'll try and get to it. unfortunately you're not the first to ask.

- anonymous

dont take james away form me:(

- anonymous

i can wait

- JamesJ

quick rld613. I want to move on with you to yet another example afterwards

- Hero

rld said she didn't need OS anymore

- JamesJ

so (d/dx) int_17^x sin(t^2)/t dt = ...

- anonymous

i didnt get that why is 17 to th epower of x

- JamesJ

\[ (d/dx) \int_{17}^x \sin(t^2)/t \ dt \]

- anonymous

yes

- JamesJ

evaluate. What is the that equal to.

- anonymous

y is it wrong what i showed u b4?

- anonymous

plz be nice to me and show me all the step? :D

- 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

LOL see u later :D

- JamesJ

I'm serious. And I'll be back in 5 minutes, so hurry.

- JamesJ

??

- anonymous

u did u come back already!!!

- JamesJ

What's the answer?

- anonymous

K can u walk me thru the steps cuz like i bever learnt this stuff?
YOu r so scary LOL

- anonymous

I was hoping i wld have the answer b4 u wld return

- anonymous

teh answer wld sinx/x?

- 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

iSo what is F(x)?????

- anonymous

That is where i am going wrong

- anonymous

oh i know si(x)?

- anonymous

d/dx(si(x)?

- 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

What is dF/dx here? What is it equal to?

- JamesJ

By the FTC, dF/dx = .... what?

- anonymous

You are forsure ready to kill me!!!!!!!!!!!!!!!

- anonymous

You are forsure ready to kill me!!!!!!!!!!!!!!!

- Hero

shakes head

- anonymous

dF/dx= sinx^2/x

- JamesJ

correct. Hence the derivative wrt (with respect to) x of the entire integral equals what?

- pokemon23

hey james can you help me with math?

- JamesJ

This thing ... what's it equal to?

##### 1 Attachment

- anonymous

sinx^2/x????? this is a guess

- JamesJ

it shouldn't be guess. it is exactly right.

- anonymous

oh ya !!!!!!!!

- anonymous

That is what i thought in the first place but i left out by accident the squared

- anonymous

I can be so annoying sometimes

- 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

oh u r so funny!!!

- JamesJ

Now write out the steps. This problem isn't exactly the same as the others.

- anonymous

ok give me a sec

- anonymous

\[d/dx=(\sqrt{x ^{6}+1}-\sqrt{x ^{3}+1})\]

- anonymous

This one i guessed

- anonymous

I think it was suppossed to be dF/dx

- 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

hint: you need to use the chain rule.

- anonymous

ok so let me try that

- anonymous

james u gotta help me out on this one

- 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

alrighty thanks james

- anonymous

I will call you back when i get the

- anonymous

I got the answer :D

- anonymous

\[2x \sqrt{x ^{6}+1} - \sqrt{x ^{3}+1}\]
Here is my answer.
Thanks james for your help. I really appreciate it

- JamesJ

that's it. Good

- 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

Oh this is very simple there is just one small catch that you have to differentiate it twice

- anonymous

d/dx(ln(cosx))
(1/cosx)*-sin(x)= -sin(x)/cos(x)
-tan(x)

- JamesJ

exactly, good.

- 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

oh u are here LOL

- anonymous

Thanks for helping me out

- JamesJ

sure. have a good vacation.

- anonymous

U better take a vacation from OS tooo!!! :D

- JamesJ

I might just do that.

- anonymous

Ya u need a break to refresh again

- anonymous

OS wldnt survive without James myinaya satellite and Hero

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