integration – Indefinite integral of $log(x+e^x)$

Today I came across an integral which I couldn’t solve:
$$intlog(x+e^x)dx$$
Wolfram alpha also couldn’t give the answer, but it gave this:
$$int_{-W(1)}^{0}log(x+e^x)dx=-0.60206974244…$$
(Note: I didn’t give it the limits of the integral, it automatically showed it) What is the exact value of that definite integral? And what is it’s connection with the Lambert-W function? d
These are supplementary questions. My main question is: What is the value of the indefinite integral? I think it should be in terms of some special functions.
I know I didn’t give any work. I tried to solve it, but couldn’t. Please give the whole derivation, not only the answer. If you know the answer but not the derivation, you can mention it in the comments.

numerical integration – Change Replace Rule with Function Input

I am trying to create a function that takes in an input $mu$ and place that input into the replace rule for a numerical integration, as seen in the code below:

Clear("Global`*")
x = r/R;
Subscript(U, T) = (CapitalOmega) R (x + (Mu) Sin((Psi)));
Subscript(U, R) = (CapitalOmega) R (Mu) Cos((Psi));
U = Sqrt(Subscript(U, T)^2 + Subscript(U, R)^2);
NIntegrate(
   NIntegrate(
    1/2 (Rho) U^3 Subscript(C, Subscript(d, 0)) /. #, {r, 0, 
      R} /. #), {(Psi), 0, 2 Pi} /. #) & @{R -> 1, (CapitalOmega) ->
    1, Subscript(C, Subscript(d, 0)) -> 1, (Rho) -> 1, (Mu) -> 1}
Subscript(P, 0)((Mu)_) = 
 NIntegrate(
    NIntegrate(
     1/2 (Rho) U^3 Subscript(C, Subscript(d, 0)) /. #, {r, 0, 
       R} /. #), {(Psi), 0, 2 Pi} /. #) & @{R -> 
    1, (CapitalOmega) -> 1, 
   Subscript(C, Subscript(d, 0)) -> 1, (Rho) -> 1, (Mu) -> (Mu)}
Subscript(P, 0)(1)

Running this code has the following result:

As you can see, simply subbing in for $mu$ using $mu rightarrow 1$ works fine, but it does not work when attempted as a function. How would I set up this function to correctly send the input to the replace rule?

integration – Let $f(x)$ determine the value of $g(x)$ in terms of $f(x)$

Let $f(x)$ determine the value of $g(x)$ in terms of $f(x),$ my teacher tells me that the solution is between sustitution, i attemped $u = log(x) ,du = frac{1}{x},dx$, $dx = e^u,du$, also my teacher says that the interval of integration won’t be the same but it’s possible to split the integral, sorry im not native english speaker lol

$$f(x) = int_0^x e^{e^t} , dt$$
$$g(x) =int_2^3 frac{dx}{log(x)}$$

microservices – Service integration with large amounts of data

I am trying to assess the viability of microservices/DDD for an application I am writing, for which a particular context/service needs to respond to an action completing in another context. Whilst previously I would handle this via integration events published to a message queue, I haven’t had to deal with events which could contain large amounts of data

As a generic example. Let’s say we have an Orders and Invoicing context. When an order is placed, an invoice needs to be generated and sent out.

With those bits of information I would raise an OrderPlaced event with the order information in, for example:

public class OrderPlacedEvent
{
    public Guid Id { get; }
    public List<OrderItem> Items { get; }
    public DateTime PlacedOn { get; }
}

from the Orders context, and the Invoicing context would consume this event to generate the required invoice. This seems fairly standard but all examples found are fairly small and don’t seem to address what would happen if the order has 1000+ items in the order, and it leads me to believe that maybe integration events are only intended for small pieces of information

The ‘easiest’ way would be to just use an order ID and query the orders service to get the rest of the information, but this would add coupling between the two services which the approach is trying to remove.

Is my assumption that event data should be minimal correct? if it is, how would I (or even, is it possible to?) handle such a scenario where there are large pieces of data which another context/service needs to respond to, correctly?

integration – Discard the Pettis integral over inequalities

Let $(E,leq)$ be a Partially Ordered Banach Space.

Let $X:=mathcal{C}(I,E)$ with $I:=(0,1)$.

Let $f:Itimes Erightarrow E$ be a function with $t mapsto f(t,x(t)) $ Pettis integrable
for each $xin X$.

Assume that there exists $x_1,x_2in X$ such that, $forall xin X,;forall tin I$: $$int_{0}^{t}f(s,x_1(s))ds leq int_{0}^{t}f(s,x(s))ds leq int_{0}^{t}f(s,x_2(s))ds.$$

May we discard the integral on the inequalities, and then $forall xin X,;forall tin I$: $$f(t,x_1(t))ds leq f(t,x(t))ds leq f(t,x_2(t))ds.$$

IF NOT, any counterexample is very welcomed.


N.T: I’m not very familiar with Pettis integration that deep, so my question may be not appropriate to MO.

integration – If $f(x)$ is integrable then $f(x^{alpha})$ is integrable?

Suppose $f(x)$ is integrable on $(0,1)$ and take any real number $alpha$ such that $alphaneq0$. Is it possible for $f(x^{alpha})$ to be integrable on $(0,1)$?
I was trying to figure out some particular examples to know how this works but I think I found a counterexample, if $x=frac{1}{2}$ and $alpha=-1$ then $x^{alpha}=2$ and $x^{alpha}notin(0,1)$. If I’m incorrect and $f(x^{alpha})$ is integrable, is it possible to extend this reasoning to more general abstract spaces such as $L^{p}$ ?

real analysis – Was Cantor aware of Lebesgue theory of integration?

Georg Cantor died in 1919, more than ten years after appearance of the Lebesgue theory of measure and integration at the beginning of the twenty-first century. It is well-known that Cantor suffered from mental illness in his last years. Lebesgue theory has a deep connection with Cantor’s theory of sets, for instance one of first Lebesgue’s contributions after his thesis was about Fourier series, which is one of motivations of Cantor in developing theory of sets. It seems interesting to know about any (possible) reaction of Cantor to the measure and integration theory of Lebesgue.

python – solve and integration differential problem c++

I want to write c++ code that first numerically compute dy/dt=f(y,t) and then after that evaluate
integration as form g(t)=h(t)+y(t)…that y is solution of ode that introduce as dy/dt=f(y,t) and h is function of t for example h(t)=t.
and bound for integration is a and b. integration(g(t),a,b)

integration – Extending Gauge integral to higher dimensions/spaces and analog of Riemann rearrangement theorem for it

The Gauge/Henstock-Kurzweil integral allows for the integration of a very large set of functions in R, at the cost of many of the nice properties of Lebesgue integration, of which it is a strict generalization in R. Essentially, it is a “conditionally convergent” version of Lebesgue integration, where functions that are not absolutely integrable can still be systematically assigned values without defining them as “improper” integrals.

For series, jettisoning absolute convergence yields the Riemann rearrangement theorem where the order of the sequence being summed matters when assigning a value to the sum (if it exists).

Are there any reasonable extensions of the Gauge integral that eschew some of the order-independent niceties (Fubini’s theorem for one)? Have these any use beyond theoretical niceties?

integration – Calculate the Euler-Lagrange for a functional with two nested integrals?

I’ve been reading papers about a fairly unknown topic in quantum mechanics called the quantum backflow effect. And in many of the papers they find an eigen value problem corresponding to the maximal amount of backflow. I understand all of the proof except from the last step in which they find the Euler-Lagrange of a functional ($I(phi)$) to maximise it.

$$I(phi) = int^infty_0{int^infty_0{phi^*(p)K(p,q)phi(q)dp}dq} – lambdaint^infty_0{phi^*(p)phi(p)dp}$$

Is the functional to be maximised and the Euler-Lagrange is

$$int^infty_0{K(p,q)phi(q)dq} = lambdaphi(p).$$

Although none of the papers explain why this is the case and I’ve been searching the internet for how to solve a problem in this form for days and can’t seem to find anything. If anyone knows why this is the Euler-Lagrange of $I(phi)$ any help would be greatly appreciated. Thanks in advance.