# Any insights on this equation?

Hi. Sorry I couldn't think of a more appropriate title for this thread.

I'm doing some calculatons and I arrived at an equation with this form
$$f(t')|_0^t = \int_0^t f(t')dt'$$

I was just wondering if anyone as any insight into the interpretation of such an expression. All I can think of is the obvious: that the difference between the function f(t) at the endpoints must equal the area under the curve f(t) between those endpoints. I guess what I'm asking is, what characteristics must f(t) have in order to satisfy this.

Moreover, and probably importantly, I only need this to hold from SOME value of t, not ALL values of t.

Thanks.

Well it looks to me like f(t') = f'(t')

What function is it's own antiderivative? Maybe do without the prime in there cus' that's just confusing. Write it as:

$$f(x)\biggr|_0^t=\int_0^t f(u)du$$

The function f(x) = e^x would do the trick, but then the equation is satisfied for ALL t. I'm particularly interested in cases where the equation is satisfied for discrete values of t.

Think of it like this: The left hand side and right hand side are DIFFERENT functions (e.g. g(t) and h(t)). Under what conditions would these functions intersect at some value of t?

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$$e^{x}f(x)|_0^t = \int_0^t e^uf(u)du$$