- #1

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## Main Question or Discussion Point

Hi pf!

I'm wondering how to evaluate. $$\int_{x_i}^x \int_{x_i}^x (ds)^2$$ I would do it like $$\int_{x_i}^x \int_{x_i}^x (ds)^2 \\ =\int_{x_i}^x ds \int_{x_i}^x ds \\= (x-x_i)^2$$ yet i know this is wrong since the answer should be ##(x-x_i)^2/2!## (taylor series is the application here). It looks like we should evaluate this as $$\int_{x_i}^x \int_{x_i}^x (ds)^2 = \int_{x_i}^x s (ds) = s^2/2$$ and then suddenly place the ##x-x_i## inside the ##s## term (which we obviously don't normally do).

Thanks so much!

I'm wondering how to evaluate. $$\int_{x_i}^x \int_{x_i}^x (ds)^2$$ I would do it like $$\int_{x_i}^x \int_{x_i}^x (ds)^2 \\ =\int_{x_i}^x ds \int_{x_i}^x ds \\= (x-x_i)^2$$ yet i know this is wrong since the answer should be ##(x-x_i)^2/2!## (taylor series is the application here). It looks like we should evaluate this as $$\int_{x_i}^x \int_{x_i}^x (ds)^2 = \int_{x_i}^x s (ds) = s^2/2$$ and then suddenly place the ##x-x_i## inside the ##s## term (which we obviously don't normally do).

Thanks so much!