Question

about the fundamental theorem of calculus.
1 question idk, pliz.

Answer 1

\[\large h(x)=\int\limits_{1}^{e^x}\ln(t)~dt\]how would I find the, \[{\large h'(x)}~~~~~~?\]

Answer 2

it would be obvious with an upper limit of x, but how do I go about it, when I have e^x as my upper limit?

Answer 3

lets see, as I am waiting, I will extract the answer using normal math, and then will see how it is done with a fund. trm of calc

\[\int\limits^{e^x}_{1}\ln(t)dt=t \ln(t)-t+C \Big |^{e^x}_{1}\]\[=e^x \ln(e^x)-e^x...\]\[=xe^x-e^x...\]\[=e^x(x-1)\]
the derivative,
\[e^x(x-1)+e^x\]\[xe^x\]

Answer 4

@dan815 @DanJS @MDoodler please help em do it using the fundamental theorem of calculus ?

Answer 5

\[\frac{d}{dx}(\int\limits_{a(x)}^{b(x)}f(t)dt ) \\ \frac{d}{dx}(F(t)|_{a(x)}^{b(x)}) \\ \frac{d}{dx}(F(b(x))-F(a(x)) \\ \frac{d}{dx}F(b(x))-\frac{d}{dx}F(a(x)) \\ b'(x)F'(b(x))-a'(x)F'(a(x)) \\ b'(x)f(b(x))-a'(x)f(a(x))\]
assuming f is continuous on [a(x),b(x)] and differentiable (a(x),b(x))
and where F'=f

Answer 6

k, when the a(x) is 1, then it would be just b'(x)f(b(x)) (the f'(a) will be all zeros)

b(x) is e^x in this case, so then it would be, e^xln(e^x) , (then exponent goes down, ik, but is this the correct set up ?)

Answer 7

(only this time, because e^x ' = e^x )

Answer 8

\[b'(x)f(b(x)) \\ b(x)=e^x \text{ and } f(t)=\ln(t) \\ \]
yep you are right
b'f(b)=e^xln(e^x)

Answer 9

or e^x *x(ln(e))

Answer 10

same answer:D

Answer 11

or e^x*x

Answer 12

ty

Answer 13

I got another similar like this, I will tag you in a new question when I am done, if you don't mind.

Answer 14

also did you ever get the integration by parts question answered like you wanted?

Answer 15

which one?

Answer 16

http://openstudy.com/users/idku#/updates/54b44e1ae4b04fb91fd8a55d

Answer 17

Yes, tnx for mentioning that, I see it

Answer 18

I guess that is fine, tnx. I will finish my h/w and when I got time, I will be back to this stuff that I do on my own.

Answer 19

ty for the help 1s again

Answer 20

ok you can mention me in your next question if you want

Answer 21

ty