Suppose that f is a continuous function on R and define a function F on [0, infinity) by f(x)= the integral of f from x to -x. Find a simple expression for F'(x).

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- Nov 7th 2009, 02:30 PMfriday616analysis help
Suppose that f is a continuous function on R and define a function F on [0, infinity) by f(x)= the integral of f from x to -x. Find a simple expression for F'(x).

- Nov 7th 2009, 03:21 PMtonio

So $\displaystyle F(x):=\int\limits_{-x}^xf(t)\,dt$

Let us take a primitive of f: $\displaystyle G'(t)=f(t)\,\,\forall,t$. By the Fundamental theorem of Integral Calculus we get that $\displaystyle \int\limits_a^bf(t)\,dt=G(b)-G(a)\,,\,\,\forall\,a\,,\,b$, so:

$\displaystyle F(x)=G(x)-G(-x)\Longrightarrow \,F'(x)=G'(x)+G'(-x)=$...

Tonio