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$$ \int_0^{10^{50}}\sin(10^{20} \sqrt{x^2 + 10^{100}})dx$$

The interval is large and I have problem seeing how to calculate it with the common numerical integral methods as they require taking lot of points across the function.

This differs from the other similar questions as I'm also considering the case of SinASinB, which leaves a function like $$\int_0^{10^{50}}\cos((10^{20} \sqrt{x^2 + 10^{100}}) - (10^{13} \sqrt{x^2 + 10^{100}}))dx $$ to be integrated.

Smithy
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  • Replacing $x$ by $10^{50}y$ reduces some of the craziness... – paul garrett Aug 04 '22 at 18:28
  • @paulgarrett I don't quite follow do you mean $$\int_0^1 sin(10^{20} \sqrt{10^{50}y + 10^{100}}) dx$$ but that will just shrink the function, the points I will need to compute won't necessarily change will they? – Smithy Aug 04 '22 at 18:41
  • Well, it'd be $10^{50}\cdot\int_0^1 \sin(10^{20+50}\cdot \sqrt{y^2+1});dy$ – paul garrett Aug 04 '22 at 18:43
  • You bloody genius it worked lol, now I can actually tackle similar problems like these, I presume taking similar steps will work even if the number in the square bracket is not a multiple of 10? – Smithy Aug 04 '22 at 18:57
  • @paulgarrett what about if I have a function like this? $$ \int_0^{10^{50}}\sin(10^{20} \sqrt{x^2 + 10^{100}}) \sin(10^{13} \sqrt{x^2 + 10^{100}})dx$$ – Smithy Aug 04 '22 at 19:18
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    @paulgarrett this simplifies to SinASinB, which leaves a function like $$ \int_0^{10^{50}}\cos((10^{20} \sqrt{x^2 + 10^{100}}) - (10^{13} \sqrt{x^2 + 10^{100}}))dx$$ behind. – Smithy Aug 04 '22 at 19:29

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