# Numerical integration with large initial limit

**URL:** <https://community.wolfram.com/t/numerical-integration-with-large-initial-limit/23023>\
**Category:** Questions\
**Tags:** calculus, numerical-computation\
**Created:** [October 27, 2023, 1:34pm UTC](https://community.wolfram.com/t/numerical-integration-with-large-initial-limit/23023 "2023-10-27T13:34:02Z")\
**Posts on this page:** 7\
**Page:** 1

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**Author:** ![safinazsalem](https://avatars.discourse-cdn.com/v4/letter/s/779978/32.png) [@safinazsalem](https://community.wolfram.com/u/safinazsalem)\
**Post date:** [October 27, 2023, 1:34pm UTC](https://community.wolfram.com/t/numerical-integration-with-large-initial-limit/23023/1 "2023-10-27T13:34:02Z")

</div>

I’m facing an issue with the integral attached in the MA file.

Can Mathematica establish this integral with large integration initial limit (a =- 10^9) ? In the current MA file, the integration only works because I set the initial integration limit to (a=1).

I’m using MA 12:00, so can integration Method used or so?

Also, how to take the limit of the integration at q \to 0 ?

I mean one can take the limit of Integrate like:

```
Limit[Integrate[Exp[a x], {x, 1, q}], q -> 0]

```

but how to do so with NIntegrate ?

Thanks,  
Safinaz Salem

[Open in Wolfram Cloud ↗](https://www.wolframcloud.com/obj/dd3d2cad-5e74-4888-ae2c-fb796c39a609)

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**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\
**Post date:** [October 27, 2023, 2:16pm UTC](https://community.wolfram.com/t/numerical-integration-with-large-initial-limit/23023/2 "2023-10-27T14:16:29Z")

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The integral `i1` can be done symbolically:

```
Map[Integrate[#, {b, 1, q}] &, eq[k, q, b, f] //
   FullSimplify // Expand]

```

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<div class="post-metadata">

**Author:** ![safinazsalem](https://avatars.discourse-cdn.com/v4/letter/s/779978/32.png) [@safinazsalem](https://community.wolfram.com/u/safinazsalem)\
**Post date:** [October 27, 2023, 7:11pm UTC](https://community.wolfram.com/t/numerical-integration-with-large-initial-limit/23023/3 "2023-10-27T19:11:17Z")

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Hello, thanks a lot for your answer. You mean to try something like:

```
i1[k_, f_,q_]=Map[Integrate[#, {b, a, q}] &, eq[k, q, b, f] // FullSimplify // Expand]

i2[k_,q_]=Map[Integrate[#, {f, a, q}] &, i1[k,f,q] // FullSimplify // Expand]

```

But it takes so much time to be evaluated. Is there a way to speed up the integration ?

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<div class="post-metadata">

**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\
**Post date:** [October 27, 2023, 9:07pm UTC](https://community.wolfram.com/t/numerical-integration-with-large-initial-limit/23023/4 "2023-10-27T21:07:44Z")

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I only tried the `i1`. The `i2` is probably too difficult for a symbolic solution.

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<div class="post-metadata">

**Author:** ![safinazsalem](https://avatars.discourse-cdn.com/v4/letter/s/779978/32.png) [@safinazsalem](https://community.wolfram.com/u/safinazsalem)\
**Post date:** [October 27, 2023, 9:33pm UTC](https://community.wolfram.com/t/numerical-integration-with-large-initial-limit/23023/5 "2023-10-27T21:33:30Z")

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I try to evaluate i2[k,q] in the limit where q\to0, so I guess to take The limit, i2 should be done by Integrate too, because how to take the limit in NIntegrate?

In the attached MA file, I evaluate i2[k, 0.01] to consider q is so small, but I don’t think this is accurate and one should take the Limit.

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<div class="post-metadata">

**Author:** ![gianlucagorni](https://avatars.discourse-cdn.com/v4/letter/g/ec9cab/32.png) [@gianlucagorni](https://community.wolfram.com/u/gianlucagorni)\
**Post date:** [October 28, 2023, 8:58am UTC](https://community.wolfram.com/t/numerical-integration-with-large-initial-limit/23023/6 "2023-10-28T08:58:38Z")

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Your `i2` can be written as a double integral on a variable square. Make a change of variable so that the integral become on a fixed square. Then find the limit of the function inside the interval, if possible. That may be a start.

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<div class="post-metadata">

**Author:** ![safinazsalem](https://avatars.discourse-cdn.com/v4/letter/s/779978/32.png) [@safinazsalem](https://community.wolfram.com/u/safinazsalem)\
**Post date:** [October 30, 2023, 6:50am UTC](https://community.wolfram.com/t/numerical-integration-with-large-initial-limit/23023/7 "2023-10-30T06:50:25Z")

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Thanks a lot I will give it a try.