the function of interest

**Objective:**The objective is to minimize the Φ function to calculate E (activation energy) for a given α. This E is calculated as the local minima between -500 and 500.Properties of the function

• The i and j are heating rates. They are both [5 7 10 12 15 20].
• Inside the integral, E and R are constant. R is the universal gas constant 8.314. But, E is unknown.
• Upper limits and lower limits of T are known. Means there is another function feeding in the values to it.

What the literature suggests:

Literature suggests to evaluate the integrals in the numerator and denominator with the known values of T and R using trapezoidal method. Then, to substitute this symbolic expression inside the summation and minimize it (between -500 and 500) to calculate the unknown value of E.

The Problem:

The problem is that E is unknown, I could not figure out how to use trapezoidal rule. Because it becomes a symbolic integration. I tried using the "int" method. But the result contains the exponential integral (ei). Substituting this result in the summation leads to wrong values of E.The function Φ is a non-linear function. But, my code results in a simple linear function mainly because it is neglecting the ei part of the integral.

My Question:

is there a method to symbolically solve the Φ function using trapezoidal or any other method with E as the unknown?

ps: this function is part of a thermal kinetics research problem of a solid phase amorphous susbtance under elevated heating programs. Hence, i have no else to ask but the community :)

UPDATE:

It has been pointed out about the problem of T. I will rephrase the question with a simple example that isolates the issue instead of showing a complicated formula.. Sorry.

How to integrate the function exp (-E/(8.314*T) between the T values of lower limit = 300 and upper limit = 310

• E is kept constant.
• the integration will definitely result in an exponential integral because this kind of integration doesn't have a unique solution in maths. So, can we apply trapezoidal method for this symbolic integration?

If the above can be done, I think remaining is easy. It's just summation and minimization of one variable.

How would you solve this integral?

http://www.hostmath.com/Show.aspx?Code=%5Cint_%7B-%5Cinfty%7D%5E%7B%5Cinfty%7D%20e%5E%7B-i(Ax%2BBx%5E%7B2%7D)%7Ddx

I did this completing the square, but is there a better way to solve this?

Hey!

I'm trying to find the probability that a state |0> in a 1-dimensional harmonic oscillator will be excited to the state |1> at t=infinity if there is a pertubation to the potential H' = constant * x * exp(t^2 / tau^2 ) at t=-infinity.

When I write the expression for the amplitude c_|1> I end up with an integral: new constant* integral[ exp(-(t^2 / tau^2 - i omega t)) dt] with the boundaries going from t=-infinity to t=infinity which I don't know how to solve.

I know the solution if the exponential had been real, but I'm not sure what to do now there is an imaginary part - and Wolfram Alpha isn't very helpful.

EDIT: Apparently I can't make the latex-thing work either.

Hi, I was trying to integrate the exponential form of tangent earlier today when I noticed something weird:

I expected to find that it equals -ln(|(e⁻ᶦˣ + eᶦˣ)/2|) + C, which is -ln(|cos(x)|) + C, but instead, I got -ln(|e⁻ᶦˣ + eᶦˣ|) + C, which is -ln(|2*cos(x)|). I checked my math on Symbolab and that gave me the same thing.

Would anyone be willing to try to do this integral to see if you get anything else? Thank you so much.

I have an integral that works, but I'm not sure how to justify it

I have the integral in the form

\int exp(s)cos(st)(1,0)^T - \int exp(s)sin(st)(1,0)^T

Then if I instead change (1,0)^T to 1, and (0,1)^T to i, I have

\int exp(s)cos(st) - \int i * exp(s)sin(st)(1,0)^T

Which can be expressed as

\int exp(s)(cos(st) - i * sin(st))

Then eulers gives

\int exp(s)(exp(-ist))

And

\int exp(s(1 - it))

Which is straight forward...

Once I've computed the exponential integral I can factor things out to the form

Re(foo) + i Im(bar)

Then, I can sub back for i a

foo (1, 0)^T + bar(0,1)^T

But I'm not sure how I should explain the reasoning for this?

Currently I'm just pointing at R^2 and C and saying that there's a bijection between the two, so that it "should" work out fine (and it does).

So

The crux of my question is; how to explain going from

\int exp(s)cos(st) - \int i * exp(s)sin(st)(1,0)^T

to

\int exp(s)(cos(st) - i * sin(st))

Then integrating as

\int exp(s(1 - it))

Then factoring and switching back again.

It works, i just don't know how to explain why.

Thanks

So I'm taking AP Calc BC this year, and several months back we started looking at anti-derivatives and integrals. We learned that Ln[x] is defined as the integral of 1/t dt from 1 to x. We then saw by proof that the inverse of Ln, Exp, is its own derivative. Thereafter we chugged along the curriculum, and my teacher, who is normally pretty good about justifying his statements, surreptitiously switched the notation Exp[x] to e^(x). I shrugged it off.

Months later, I'm realizing his jiggery-pokery. I confronted him about it and he could not answer. So my question to you, reddit, is this: Why is Exp (again, defined to be the inverse of the integral of 1/x) expressible in the form a^x (ie an exponential function, I'm not really concerned about the base being e). Equivalently, why is the integral of 1/x expressible in the form Log*a*[x]?

Heuristically, it makes sense to me that Exp, having the properties of exponentials (distributivity over multiplication, for instance), is an exponential itself. But I'd like to see a proof. Thank you!

I understand the following formula; http://gyazo.com/f688b14f2affc2e7ba09b0ad0698326f

But in this example; http://gyazo.com/a63a6f4b0655a76cdf460ccfe9f68076

Were has the 1/2 come from infront of the e please ?

Thanks

So I'm writing about a salary situation in which I must choose an option of three functions for a salary for 35 years. The three options are:

f(x)= .01*2^x

f(x)= 10,000,000

f(x)=2,000,000 (1+ .05/2)^2x

I would like to include a section in which I talk about the total salary after 35 years, not just the money earned in the last year (which is the method of evaluating that most of my classmates are using). I understand how to take definite integrals of polynomial functions, and am able to calculate the area under the relevant parts of the curves using my calculator, but I would like to understand how these values are derived, for explanation purposes as well as for future reference (I'm taking AP Calc BC next year). Can anybody shed some light on the subject?

I'm doing a physics assignment and I got stuck on this integral I need to solve.

The integral I need to solve is this one (there are other constants in front but I suppressed them)

[; I_2=\frac{1}{r}\int_0^{\infty} e^{ir\sqrt{p^2+i\hbar 2mu}/\hbar}du ;]

I was given that the value is

[; I_2=\frac{1}{r^2}(p+\frac{i \hbar}{r})e^{ipr/\hbar} ;]

But I'm stumped as to how to get this result. It's possible that my integrand isn't correct, but I'm about 90-95% sure it's right.

edit: It appears my latex isn't being displayed by my tex the world plugin, gonna work on that.

I'm a hydrogeology student and I'm kinda stuck trying to go backwards on an equation I found in a textbook. Could someone help me with working backwards in an exponential integral please?

I want to isolate x in: y=Ei((r^2)/(4xk))

I am trying to encode the Ei(x) function into JMP in order to do some nonlinear modeling. There is no Ei(x) function by default in JMP, but there is a partial gamma function.

Fortunately, as this page demonstrates there is a very direct relationship between the upper incomplete gamma function and the exponential integral function. Unfortunately, the incomplete gamma function that is in JMP is the lower incomplete gamma function. I've tried to figure out how to relate the lower incomplete gamma function to the exponential integral function, but I am afraid it is out of my reach. Can anyone out there give me some help?

Edit: I should point out that I will only be dealing with real, positive x values.