How to find the first three terms of the Taylor series around $b=c$ of
$$ \int^{\frac{1}{b}}_{a}f(b,x) dx $$
2026-04-12 23:12:07.1776035527
Is it possible to find the Taylor series of an integral where the upper limit depends on the variable?
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Assuming you mean the taylor series of the function $F$ where $$ F(b)=\int^{\frac{1}{b}}_{a}f(b,x) dx $$ Let's find the coefficients of the Taylor Series.
We know by Liebniz Rule that, assuming $f$ has a continuous partial derivative with respect to $b$ (I believe this is necessary and sufficient, please correct me if I am wrong), $$ F'(b)=\int^{\frac{1}{b}}_{a}\frac{\partial f}{\partial b}(b,x) dx-\frac{1}{b^2}f(b,1/b) $$ And similarly for the second and third order derivatives. Then plug in $c$ for your coefficients.