## theming – How do I integrate a custom header and footer?

I need to add a custom header to a theme that I am currently building.

The only way I can seem to get the header and footer to show up on the page is to hard code it into the page.tpl.php file. I’ve tried to use the include trick, but that isn’t working.

The header and the footer need to appear on every page.

Any help?

## 9 – Integrate webform submissions with Google sheet

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## How to integrate images into text unity?

Here are two screenshots from Stellaris where images are seamlessly integrated into the text. How can I achieve this in unity?  ## calculus and analysis – How to integrate faster

I always wanted to know how to speed up my integral computation in Mathematica. Is there some techniques that i am unaware to make the integration faster.

Here an example:

``````func = (2 (μG2-μPi2) (5 x^2+5 x (z-2)-4 (η-ρ+z-1)))/mb^2-12 (x+z-2) (-η-ρ+x+z-1)
``````

The integration is given as:

``````Integrate(func, {x, 2*Sqrt(η), 1 + η - ρ},
{z, -((-(2*(η - ρ + Sqrt(x^2 - 4*η) - 1)) + x*(Sqrt(x^2 - 4*η) - 2) + x^2)/
(Sqrt(x^2 - 4*η) + x - 2)), (2*(-η + ρ + Sqrt(x^2 - 4*η) + 1) -
x*(Sqrt(x^2 - 4*η) + 2) + x^2)/(Sqrt(x^2 - 4*η) - x + 2)},
Assumptions -> {0 < ρ < 1, 0 < η < 1, ρ < η, η + 1 > 2*Sqrt(η) + ρ,
Element(mb, Reals)}, GenerateConditions -> False)
``````

On my laptop i need 42.219 s to solve the integral. However, my integrals are getting more and more complicated so to learn new optimization methods would be much appreciated.

## calculus and analysis – Integrate real function returns complex value

I want to compute the integral
$$int_0^c exp(-cx+x^2) mathrm{d}x,$$
where $$c>0$$ is an unknown constant. In Mathematica Version 12.2.0

``````\$Version

(* 12.2.0 for Microsoft Windows (64-bit) (December 12, 2020) *)
``````

I evaluate the integral and I obtain

``````Integrate(Exp(-c*x + x^2), {x, 0, c}, Assumptions -> {c > 0})

(* E^(-(c^2/4)) Sqrt((Pi)) Erfi(c/2) *)
``````

Why do I obtain a complex expression? Is it a bug or I am doing something wrong?

## Integrate conjugate transpose

I can’t find the

Integrate[conjugate Transpose[u[x, t]] *D[u[x, t], {x, 1}],x] where u is complex function and u.conjugate[u] =||u||^2

Thanks

## numerical integration – How to define a polygonal region in 2D to subsequently integrate over it?

Here is an example in 12.2.

``````poly = Polygon({{0, 0}, {1/2, Sqrt(3)/2}, {1, 1/Sqrt(3)}, {1, 0}});
NIntegrate(Log(x + y + 1), {x, y} (Element) poly)
``````

`0.366623`

Let us verify it by

``````Integrate(Log(x + y + 1), {x, y} (Element) poly)
``````

`-((36 - 12 Sqrt(3) + 12 Log(2) + 228 Sqrt(3) Log(2) + 138 Log(3) + 54 Sqrt(3) Log(3) + 9 Log(4) - 3 Sqrt(3) Log(4) + 48 Sqrt(3) Log(6) - 2 Log(8) - 2 Sqrt(3) Log(8) + 2 Sqrt(3) Log(9) - 48 Sqrt(3) Log(2 - 2/Sqrt(3)) + 90 Log(2 - Sqrt(3)) + 54 Sqrt(3) Log(2 - Sqrt(3)) - 90 Log(3 - Sqrt(3)) - 54 Sqrt(3) Log(3 - Sqrt(3)) - 180 Log(-1 + Sqrt(3)) - 108 Sqrt(3) Log(-1 + Sqrt(3)) + 72 Log(1 + Sqrt(3)) - 48 Sqrt(3) Log(1 + Sqrt(3)) - 36 Log(2 + Sqrt(3)) + 24 Sqrt(3) Log(2 + Sqrt(3)) - 18 Log(3 + Sqrt(3)) - 90 Sqrt(3) Log(3 + Sqrt(3)) - 48 Log(6 + Sqrt(3)) - 52 Sqrt(3) Log(6 + Sqrt(3)) - 72 Log(3 + 2 Sqrt(3)) + 48 Sqrt(3) Log(3 + 2 Sqrt(3)) + 36 Log(9 + 5 Sqrt(3)) - 24 Sqrt(3) Log(9 + 5 Sqrt(3)))/(8 Sqrt( 3) (19 + 11 Sqrt(3)) (-45 + 26 Sqrt(3))))`

``````N(%)
``````

`0.366623`

Addition. `NIntegrate` produces a different result if the vertices are taken couunter-clockwise as

``````poly1 = Polygon({{1, 1/Sqrt(3)}, {1/2, Sqrt(3)/2}, {0, 0}, {1, 0}});
NIntegrate(Log(x + y + 1), {x, y} (Element) poly1)
``````

`0.17812`

shows. `Integrate` produces the same:

``````Integrate(Log(x + y + 1), {x, y} (Element) poly1)
``````

`(-18 - 18 Sqrt(3) + 117 Log(2) + 88 Sqrt(3) Log(2) + 297 Log(3) + 141 Sqrt(3) Log(3) - 417 Log(4) - 209 Sqrt(3) Log(4) - 574 Log(8) - 287 Sqrt(3) Log(8) - 396 Log(27) - 228 Sqrt(3) Log(27) + Log(216) + Sqrt(3) Log(216) + 6 Log(1728) + 4 Sqrt(3) Log(1728) - 6 Log(46656) - 4 Sqrt(3) Log(46656) + 4 Log(452984832) + 2 Sqrt(3) Log(452984832) - 594 Log(18 - 8 Sqrt(3)) - 342 Sqrt(3) Log(18 - 8 Sqrt(3)) - 288 Log(11 - 5 Sqrt(3)) - 144 Sqrt(3) Log(11 - 5 Sqrt(3)) + 594 Log(9 - 3 Sqrt(3)) + 342 Sqrt(3) Log(9 - 3 Sqrt(3)) + 1188 Log(5 - Sqrt(3)) + 684 Sqrt(3) Log(5 - Sqrt(3)) + 288 Log(-8 (-2 + Sqrt(3))) + 144 Sqrt(3) Log(-8 (-2 + Sqrt(3))) - 591 Log(6 + Sqrt(3)) - 279 Sqrt(3) Log(6 + Sqrt(3)) + 1179 Log(7 + Sqrt(3)) + 567 Sqrt(3) Log(7 + Sqrt(3)) + 297 Log(15 + 8 Sqrt(3)) + 141 Sqrt(3) Log(15 + 8 Sqrt(3)) - 297 Log(17 + 9 Sqrt(3)) - 141 Sqrt(3) Log(17 + 9 Sqrt(3)))/(8 Sqrt( 3) (-3 + 2 Sqrt(3)) (9 + 5 Sqrt(3)))`

``````N(%)
``````

`0.17812`

## seo – Can I integrate an exiting blog platform into my website

It is likely technically doable but maybe not a great idea.

Doing this would require a reverse proxy to unify 2 different namespaces into one. (You might be able to pull it off with mod_proxy for Apache for example). This will cause undue delays and puts you at the mercy of changes by the blog provider. You may also run afoul of the remote provider t&c’s

You might want to consider blogging on your own platform and syndicating your content to other providers or using a CMS that will allow you to integrate with them using an API (assuming they offer one)

Have you looked at running WordPress for you blog? Its pretty good at that and is widely supported. Another thought is to consider putting your nlog on blog.example.com so that this can be on seperate infrastructure to your main website.

## development – Approaches to integrate our Service Desk system with SharePoint Lists

Your approach is valid, but bear in mind that the OOB Microsoft’s CSOM library will not work with SharePoint online if the legacy authentication is not allowed. So, instead, it’s best to use PnP Core library (deprecated) or the new cross platform version: PnP Framework. You can still use CSOM syntax using these libraries. But remember, that with the modern authentication, you will not be able to use login and password. Instead, you will have to rely on App ID and App secret. To get those, you will either need to register SharePoint-only app or an Azure App registration.

SharePointPnPCoreOnline Nuget Package Option 1. Granting access using SharePoint App-Only

https://docs.microsoft.com/en-us/sharepoint/dev/solution-guidance/security-apponly-azureacs

Option 2. Granting access via Azure AD App-Only

Suppose $$N$$ integral, $$Ngeq |u|$$, prove that
$$lim_{Nrightarrowinfty}int_{|z|=N+frac{1}{2}}frac{picotpi z}{(u+z)^2}dz =0.$$
$$|int_{|z|=N+frac{1}{2}}f(z)dz|leqfrac{2pi^2(N+frac{1}{2})}{{(N+frac{1}{2}-|u|})^2}sup_{|z|=N+frac{1}{2}}|cot z|.$$
If we can prove that $$sup_{|z|=N+frac{1}{2}}|cot z|=o(N) (Nrightarrow infty)$$, then we complete the proof. But I stuck here. And I want to use the result and the residue formula to prove an indentity, so the formula is of no use here. Appreciate any help!