Numerical integration of a function

For a particular physical system, the force is given by $ $ F(r)=-\frac{d\Phi(r)}{dr}=\frac{r^5+2ar^3+a^2r-4a}{r^4(r^3+ar+2a)}$ $ where $ \Phi(r)$ is the potential energy which can be obtained as the integral of $ F(r)$ . However, the integral is very complicated to solve as the cubic polynomial in the denominator of the above equation has complex roots (refer this question in Math.SE). Actually I need to obtain the variation of $ \Phi(r)$ as a function of $ r$ , i.e., the $ \Phi(r)-r$ plot.

As the evaluation of the integral is complicated, I am trying to obtain the integral curves of $ F(r)$ using the dataset of the $ F(r)-r$ plot. In this sense, I need to integrate a dataset. However, I am not sure whether this is possible.

Question: Is it possible to integrate a dataset to obtain the integral curve of a function?

Who would I need to hire for this? Website that supports upload feature & google docs integration?

I’m not sure if this is the appropriate place to ask these questions so feel free to remove this if it violates any guidelines. I’ve got no experience in web design/development or anything so I’m a bit lost here.

I want to build a website that has an upload feature where people can upload docs and those docs could be uploaded instantaneously into a platform like google docs that allows annotations. Of course, I want it to look pretty as well.

So for this, do I have to hire a separate web designer & developer and if not them who else could do something like this?

Thanks for any answers!

Integration of a peaked multidimensional function

I have the following function

PQ=p^2 + q^2 + μ - 2*p*q*(Cos[θp]*Cos[θq] + Cos[ϕ]*Sin[θp]*Sin[θq]) QK=k^2 + q^2 + μ - 2*k*q*Cos[θq] KPQ=k^2 + p^2 + q^2 + 2*p*Cos[θp]*(k - q*Cos[θq]) - 2*q*(k*Cos[θq] +    p*Cos[ϕ]*Sin[θp]*Sin[θq])  vol = p^2*q^2*Sin[θp]*Sin[θq];  denFreq = ω + (k^2 + q^2 + KPQ)/2;  f[ω_, η_, k_, p_, q_, θp_, θq_, ϕ_, μ_] = vol*(1/PQ - 1/QK)*   ((HeavisideTheta[kF - Sqrt[KPQ]]*denFreq)/(PQ*(denFreq^2 + η^2))) 

where I introduced $ \mu$ to regularize the divergence of the integrand. I should check whether the integral converge to a finite limit when $ \mu\rightarrow 0$ .

So I tried to compute the following integral (for small $ \mu$ )

kF = 1.;  ω0 = 0;  η0 = 0.1;  μ0 = 0.0001;  k0 = 0.5;  kcut = 10;  NIntegrate[f[ω0, η0, k0, p, q, θp, θq, ϕ, μ0], {p, kF, kcut},    {q, 0, kF}, {θp, 0, Pi}, {θq, 0, Pi}, {ϕ, 0, 2*Pi},    Method -> "MonteCarlo"] 

I tried different integration schemes (mainly different montecarlo schemes, including the Cuba library) but I obtain quite different results, since I guess even after the regularization with $ \mu$ the function is not “well-behaved”.

Which integration scheme is the most recommended for my case? Taking smaller values of $ \mu$ I find that the integral increases, but how can I be sure that it is the “real” behaviour of the function and not simply a consequence of a not converged numerical integration?

Any suggestion is welcome

Ahrefs api integration

Ahrefs has API plans, but they require additional subscription that starts at $ 500 / month. Unless a software developer integrates the API in his software, we can use the API with normal paid accounts, take a look https://ahrefs.com/api/integrations

Sharepoint online/365 integration (Upload files) on React app hosted on Azure and WebApi C# .Net Core 2.2

I’m struggling since some weeks ago trying to interact/automate a way to upload files from a Web App created in React and upload files to a Sharepoint Online Site – in a specific folder. The WebApp is hosted in Azure and using a C# .Net Core 2.2 as backend.

I’m trying to using some kind of REST API that help me out with this task (Could be on React in frontend, or in C# Core or C# MS FW .Net for backend) I’m searching across internet a way to do it but all the testings were failed.

Someone can give me some insight, tip or advice on how to achieve this?

I’m trying:

  • Use code from Microsoft WebPage (Using jQuery).

  • Using PnP, but on my localhost I receive a CORS problem (I’m trying using Client ID and Secret ID to interact with Sharepoint).

Integration of Interpolated function take an unacceptable amount of time

I have a simple integration which, when using an interpolation function, is taking too long to calculate:

c = 2.99792*10^5; A = 3.87624*10^-14; FreeElectronFractionData = {{3000, 1.0829044}, {2984.9246, 1.0828562}, {2969.8493, 1.0828473}, {2954.7739, 1.0828366}, {2939.6985, 1.0828238}, {2924.6231, 1.0828083}, {2909.5478, 1.0827898}, {2894.4724, 1.0827674},      {2879.397, 1.0827404}, {2864.3217, 1.0827077}, {2849.2463, 1.0826683}, {2834.1709, 1.0826207}, {2819.0955, 1.0825632}, {2804.0202, 1.0824939}, {2788.9448, 1.0824106}, {2773.8694, 1.0823111}, {2758.7941, 1.0821927},      {2743.7187, 1.0820531}, {2728.6433, 1.08189}, {2713.5679, 1.0817016}, {2698.4926, 1.0814865}, {2683.4172, 1.0812441}, {2668.3418, 1.0809745}, {2653.2664, 1.0806783}, {2638.1911, 1.0803569}, {2623.1157, 1.0800119},      {2608.0403, 1.0796454}, {2592.965, 1.0792594}, {2577.8896, 1.0788561}, {2562.8142, 1.0784377}, {2547.7388, 1.0780061}, {2532.6635, 1.0775631}, {2517.5881, 1.0771101}, {2502.5127, 1.0766486}, {2487.4374, 1.0761797},      {2472.362, 1.0757042}, {2457.2866, 1.0752228}, {2442.2112, 1.074736}, {2427.1359, 1.0742441}, {2412.0605, 1.0737472}, {2396.9851, 1.0732455}, {2381.9098, 1.0727387}, {2366.8344, 1.0722267}, {2351.759, 1.0717092},      {2336.6836, 1.0711857}, {2321.6083, 1.070656}, {2306.5329, 1.0701194}, {2291.4575, 1.0695754}, {2276.3822, 1.0690234}, {2261.3068, 1.0684627}, {2246.2314, 1.0678928}, {2231.156, 1.0673129}, {2216.0807, 1.0667222},      {2201.0053, 1.06612}, {2185.9299, 1.0655055}, {2170.8545, 1.064878}, {2155.7792, 1.0642365}, {2140.7038, 1.0635802}, {2125.6284, 1.0629083}, {2110.5531, 1.0622197}, {2095.4777, 1.0615136}, {2080.4023, 1.060789},      {2065.3269, 1.0600449}, {2050.2516, 1.0592803}, {2035.1762, 1.0584941}, {2020.1008, 1.0576853}, {2005.0255, 1.0568526}, {1989.9501, 1.0559951}, {1974.8747, 1.0551114}, {1959.7993, 1.0542004}, {1944.724, 1.0532609},      {1929.6486, 1.0522915}, {1914.5732, 1.051291}, {1899.4979, 1.0502581}, {1884.4225, 1.0491914}, {1869.3471, 1.0480894}, {1854.2717, 1.0469509}, {1839.1964, 1.0457743}, {1824.121, 1.044558}, {1809.0456, 1.0433006},      {1793.9703, 1.0420003}, {1778.8949, 1.0406552}, {1763.8195, 1.0392634}, {1748.7441, 1.0378224}, {1733.6688, 1.0363292}, {1718.5934, 1.0347802}, {1703.518, 1.0331707}, {1688.4426, 1.0314941}, {1673.3673, 1.0297415},      {1658.2919, 1.0279002}, {1643.2165, 1.025952}, {1628.1412, 1.0238707}, {1613.0658, 1.0216182}, {1597.9904, 1.0191391}, {1582.915, 1.0163525}, {1567.8397, 1.0131417}, {1552.7643, 1.0093401}, {1537.6889, 1.0047152},      {1522.6136, 0.99895508}, {1507.5382, 0.99166793}, {1492.4628, 0.9824059}, {1477.3874, 0.97072308}, {1462.3121, 0.95625418}, {1447.2367, 0.93878259}, {1432.1613, 0.91826926}, {1417.086, 0.89483803},      {1402.0106, 0.86873404}, {1386.9352, 0.84027554}, {1371.8598, 0.80981198}, {1356.7845, 0.77769325}, {1341.7091, 0.74424993}, {1326.6337, 0.70978283}, {1311.5584, 0.67455941}, {1296.483, 0.63881506},      {1281.4076, 0.60275746}, {1266.3322, 0.56657254}, {1251.2569, 0.53043096}, {1236.1815, 0.4944942}, {1221.1061, 0.45891987}, {1206.0307, 0.4238656}, {1190.9554, 0.3894916}, {1175.88, 0.35596148}, {1160.8046, 0.32344154},      {1145.7293, 0.29209849}, {1130.6539, 0.26209576}, {1115.5785, 0.23358875}, {1100.5031, 0.20671934}, {1085.4278, 0.18161021}, {1070.3524, 0.1583594}, {1055.277, 0.1370358}, {1040.2017, 0.11767584},      {1025.1263, 0.10028183}, {1010.0509, 0.084822106}, {994.97554, 0.07123293}, {979.90017, 0.05942197}, {964.8248, 0.049273035}, {949.74943, 0.040651617}, {934.67406, 0.033410795}, {919.59868, 0.027397081},      {904.52331, 0.022455872}, {889.44794, 0.018436284}, {874.37257, 0.015195242}, {859.2972, 0.012600771}, {844.22182, 0.01053444}, {829.14645, 0.008892887}, {814.07108, 0.007588385}, {798.99571, 0.006548454},      {783.92034, 0.005714659}, {768.84496, 0.005040865}, {753.76959, 0.004491226}, {738.69422, 0.004038206}, {723.61885, 0.003660784}, {708.54348, 0.003342951}, {693.46811, 0.003072499}, {678.39273, 0.002840079},      {663.31736, 0.002638491}, {648.24199, 0.002462145}, {633.16662, 0.002306669}, {618.09125, 0.00216861}, {603.01587, 0.002045214}, {587.9405, 0.00193427}, {572.86513, 0.001833981}, {557.78976, 0.001742876},      {542.71439, 0.00165974}, {527.63902, 0.001583562}, {512.56364, 0.001513494}, {497.48827, 0.001448819}, {482.4129, 0.001388928}, {467.33753, 0.001333298}, {452.26216, 0.00128148}, {437.18678, 0.001233084},      {422.11141, 0.00118777}, {407.03604, 0.001145243}, {391.96067, 0.00110524}, {376.8853, 0.001067531}, {361.80992, 0.001031911}, {346.73455, 0.000998196}, {331.65918, 0.000966223}, {316.58381, 0.000935844},      {301.50844, 0.000906924}, {286.43307, 0.00087934}, {271.35769, 0.00085298}, {256.28232, 0.00082774}, {241.20695, 0.000803522}, {226.13158, 0.000780232}, {211.05621, 0.000757781}, {195.98083, 0.00073608},      {180.90546, 0.000715041}, {165.83009, 0.00069457}, {150.75472, 0.00067457}, {135.67935, 0.000654928}, {120.60397, 0.000635516}, {105.5286, 0.000616174}, {90.453231, 0.000596693}, {75.377859, 0.000576785},      {60.302487, 0.000556011}, {45.227116, 0.000533637}, {30.151744, 0.000508209}, {15.076372, 0.000475883}, {0, 0.000410148}};   FreeElectronFraction := Interpolation[FreeElectronFractionData, InterpolationOrder -> 1]  ElectronNumberDensity[\[Eta]_] := (redShift = 6.64*^18^2/((c - Sqrt[c]*Sqrt[c - 2.*A*\[Eta]])/A)^2 - 1.; FreeElectronFraction[redShift]*1.42*^-7*(1. + redShift)^3)  Plot[NIntegrate[ElectronNumberDensity[eta], {eta, \[Eta], 3.78*^18}, MaxRecursion -> 15], {\[Eta], 1.47*^17, 2.66*^17}]  ListPlot[FreeElectronFractionData] 

The 25 seconds doesn’t seem to be a lot, but this calculation is inside another integral which didn’t complete in eight hours. As nearly as I can tell, this integral is the culprit. Specifically the interpolation function.

I’ve seen other suggested solutions on this board, but none of them worked for me. One of the solutions looked promising: creating a pure function based on the interpolated data and using that in the integral, but that is beyond my skills.

Equation solving for approximate solutions when function has numerical integration in it

I have an equation that is of one variable lat and has a numerical integration in it t from 0 to 1.

Let’s say it’s given by NIntegrate[f(lat,t),{t,0,1}]. Now, I want to solve the equation for variable lat given NIntegrate[f(lat,t),{t,0,1}] == c where c is some constant value. Solve[] clearly doesn’t work, is there a way to do this without implementing a loop guessing values?