Question

d/dx(integral from 3 to e^x of sin(t^2-3t)dt - please help with part of this problem.

Answer 1

i subbed e^x

Answer 2

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)

Answer 3

and the deriv of what is in the parens is not e^x

Answer 4

\[\int\limits\limits_{3}^{e^x}\sin((e^x)^2-3e^x)(???)\]

Answer 5

\[\frac{d}{dx} \int\limits_a^{b} f(t) dt = \frac{d}{dx} F(b) - F(a) = f(b) - f(a)\]
so we have
\[\sin(e^{2x} -3e^{x}) - \sin(3^{2} -3(3))\]
\[=\sin(e^{2x} -3e^{x})\]

Answer 6

where is the e^x out front?

Answer 7

hmm i guess i missed something

Answer 8

oh ok
\[\frac{d}{dx} F(e^{x}) - F(3) = \frac{d}{dx}e^{x} * f(e^{x}) - \frac{d}{dx}3 * f(3) = e^{x}*f(e^{x}) - 0\]

Answer 9

i forgot about chain rule :{

Answer 10

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?

Answer 11

the "e^(2x) " in sin(e^(2x) -3e^x) ?

Answer 12

yes

Answer 13

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

Answer 14

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

Answer 15

ok. so via chain rule.. why does the deriv of the inside = e^x and not 2e^2x-3e^x

Answer 16

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

Answer 17

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?

Answer 18

\[\large \int\limits_3^{e^{x}} \sin(t^{2}-3t) dt = F(e^{x}) - F(3)\]
\[\large \frac{d}{dx} (F(e^{x}) - F(3)) = (e^{x})'f(e^{x})\]

Answer 19

oooh nice!!! I see what you mean. what a trick concept.

Answer 20

umm not always but whenever a limit is constant, yes it goes away because derivative is 0

Answer 21

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

Answer 22

correct, use fund thm of calc ... just assume you know the anti-derivative already

Answer 23

omgosh.... F is anti-derivative yes?

Answer 24

yes that is standard notation, f is whatever function is given inside integral

Answer 25

I hate when i overthink everything.. now your post above makes perfect sense. Thank you for your help!

Answer 26

no problem, good luck .... yes this can be a tricky concept to grasp at first