anonymous
  • anonymous
d/dx(integral from 3 to e^x of sin(t^2-3t)dt - please help with part of this problem.
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
i subbed e^x
anonymous
  • anonymous
now i need to use chain rule.. do i multiply by the deriv of what is in the parens .. or the upper limit of integration? cause the answer is e^x sin(e^(2x)-3e^x)
anonymous
  • anonymous
and the deriv of what is in the parens is not e^x

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anonymous
  • anonymous
\[\int\limits\limits_{3}^{e^x}\sin((e^x)^2-3e^x)(???)\]
dumbcow
  • dumbcow
\[\frac{d}{dx} \int\limits_a^{b} f(t) dt = \frac{d}{dx} F(b) - F(a) = f(b) - f(a)\] so we have \[\sin(e^{2x} -3e^{x}) - \sin(3^{2} -3(3))\] \[=\sin(e^{2x} -3e^{x})\]
anonymous
  • anonymous
where is the e^x out front?
dumbcow
  • dumbcow
hmm i guess i missed something
dumbcow
  • dumbcow
oh ok \[\frac{d}{dx} F(e^{x}) - F(3) = \frac{d}{dx}e^{x} * f(e^{x}) - \frac{d}{dx}3 * f(3) = e^{x}*f(e^{x}) - 0\]
dumbcow
  • dumbcow
i forgot about chain rule :{
anonymous
  • anonymous
I guess I am still confused. when you evaluate the integral and use the chain rule.. do you drop the square in the first e^x term?
dumbcow
  • dumbcow
the "e^(2x) " in sin(e^(2x) -3e^x) ?
anonymous
  • anonymous
yes
anonymous
  • anonymous
Here is where I am confused. If I use the chain rule and multiply by the deriv of the inside, don t I get 2e^(2x)-3e^x? In the solutions our prof gave us.. he skipped a bunch of steps and did it in two steps
dumbcow
  • dumbcow
no thats stays because its "t^2" and you sub e^x for t the main thing to understand is you are not integrating, just using fundamental thm of calculus and chain rule
anonymous
  • anonymous
ok. so via chain rule.. why does the deriv of the inside = e^x and not 2e^2x-3e^x
dumbcow
  • dumbcow
ahh ok you are not taking derivative of "t^2 -3t" the inside of the sine function you are never actually taking derivative of sine function or f(t) what you are taking derivative of is the integral or anti-derivative of the sine function or F(t) we don't know what that is but it doesn't matter the chain rule part is to take derivative of what the input of F(t) or basically the limits of integral derivative of "e^x" is e^x
anonymous
  • anonymous
and since the lower limit is 3 and deriv is 0 we only use the upper limit? in these fundamental theorem of calc problems.. the bottom is always a constant number?
dumbcow
  • dumbcow
\[\large \int\limits_3^{e^{x}} \sin(t^{2}-3t) dt = F(e^{x}) - F(3)\] \[\large \frac{d}{dx} (F(e^{x}) - F(3)) = (e^{x})'f(e^{x})\]
anonymous
  • anonymous
oooh nice!!! I see what you mean. what a trick concept.
dumbcow
  • dumbcow
umm not always but whenever a limit is constant, yes it goes away because derivative is 0
anonymous
  • anonymous
so overall.. just so i know how to deal with any of these.. it is the function at the upper limit - the function at the lower limit.. but just all in one step .. not like integrating exactly
dumbcow
  • dumbcow
correct, use fund thm of calc ... just assume you know the anti-derivative already
anonymous
  • anonymous
omgosh.... F is anti-derivative yes?
dumbcow
  • dumbcow
yes that is standard notation, f is whatever function is given inside integral
anonymous
  • anonymous
I hate when i overthink everything.. now your post above makes perfect sense. Thank you for your help!
dumbcow
  • dumbcow
no problem, good luck .... yes this can be a tricky concept to grasp at first

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