Question

Solve

Answer 1

solve what?

Answer 2

\[\frac{ d }{ dx } \int\limits_{e4\sqrt{x}}^{e^(2x)} lnt dt\]

Answer 3

I cant understand how to approach this for the life of me.
Anyone? This is Calculus logarithmic differentiation

Answer 4

okay

Answer 5

To clarify the upper limit is e to the power 2x
and the lower limit is e to the power 4x^(1/2)

Answer 6

When faced with an integral that we cannot handle analytically, we can replace it by one with which we might have more success


EXAMPLE:
Using Integration by Parts
Evaluate
SOLUTION
We use the formula with
To complete the formula, we take the differential of u and find the simplest antiderivative
of cos x.
Then,
Now Try Exercise 1.
The goal of integration by parts is to go from an integral u dv that we don’t see how
to evaluate to an integral v du that we can evaluate. Keep in mind that integration by
parts does not always work.
Let’s examine the choices available for u and v in Example 1.
1
1
L x cos x dx = x sin x -
Lsin x dx = x sin x + cos x + C.
du = dx v = sin x
u = x, dv = cos x dx.
1 u dv = uv - 1v du
1x cos x dx.

Answer 7

Just to make sure I got the formula correct,

\[\frac{ d }{ dx }u*v = u \frac{ dv }{ dx } + v \frac{ du }{ dx }\]

after integrating both sides by x and rearranging we get:

\[\int\limits_{x}^{y} dv = u*v - \int\limits_{x}^{y} v du\]

did I get the first part right?

Answer 8

yes you did

Answer 9

Im not able to keep up with the example Im afraid and I by my vague understanding of what your saying I fail too see how I can apply it in my question. What is throwing my off in the question above are the upper and lower limits, To me there are variables in the limits aswell which leads me to believe that I must remove the variables from the limits somehow inorder to solve the integral

Perhaps finding the derivative of both the limits as the d/dx preceeeds the integral?

Answer 10

okay and yes finding the derivative of both limits as the d/dx proceeds the integral

Answer 11

Maybe if you draw the example so I can understand what the symbols mean with more clarity if you have the time.

Answer 12

okay

Answer 13

Ok Ill will start their then,

Answer 14

\[\frac{ d }{ dx }e ^{2x} = 2e ^{2x}\]

and

\[\frac{ d }{ dx }e ^{4\sqrt{x}} = \frac{ 2e ^{4\sqrt{x}} }{ \sqrt{x} }\]

Am I right so far?

Answer 15

Now the variables x remain in the limits and even thou by my understanding the d/dx that preceded the integral will now be gone, the limits however still leave in a position where I cannot solve the integral.

Answer 16

and yes that is correct

Answer 17

I understand the product rule however how should that apply to the question? The integral is:
\[\int\limits_{Lower Limit}^{Uppwer Limit} \ln(t) dt\]
where the upper limit = e^2x
and lower limit = e^(4x^(1/2))

Answer 18

okay hold on

Answer 19

how would I use the product rule here? I believe it has something to do with logarithmic differentiation

Answer 20

So by using integration by parts I can find
\[\int\limits \ln(t) dt = t*\ln(t) - t\]
is that correct?
After this I put in the limits right?
But the expression I get after putting in the limits is way to messy, I cant find a way to differentiate that. Any help?

Answer 21

I fear there is some simply trick to this question that I am just not seeing

Answer 22

Wow

Answer 23

What amazed you perculiar?

Answer 24

I don't think anyone can help me here, Im gonna ask my Calculus teacher how do it, ounce I understand how to do it I'll post the answer here if anyone was confused like me.

Answer 25

the derivative of the integral is the integrand, plus the chain rule
you can almost do it in your head

Answer 26

\[\frac{ d }{ dx } \int\limits_{e4\sqrt{x}}^{e^(2x)} \ln(t) dt\]

Answer 27

cant really read the upper and lower limit
is the upper limit \(e^{2x}\) ?

Answer 28

if so , one part is
\[\ln(e^{2x})\times 2e^{2x}=4xe^{2x}\]

Answer 29

Upper Limit =
\[e ^{2x}\]

Lower Limit =
\[e ^{4\sqrt{x}}\]

Answer 30

I asked my professor and he told me to use the Fundamental Theorem of Calculus and break the limits into two parts. I have a vague idea of what my professor was pointing towards but have not started working on the question. Maybe you can help me understand it better.