Channel: Quod Erat Demonstrandum

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just a so-called solution

May 29, 2014, 12:09 am

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From an old file in hard disk…

Find f(x) such that

$\int_0^x f(t)dt=f(x)-1$ .

‘solution’

Let $\int$ be “ $\int_0^x$ ” and be f(t) .

Then

$\int f=f-1$

$1=f-\int f$

$1=f(1-\int)$ 　　（factorization）

$f=(1-\int)^{-1}1$

Think about sum to infinity, we have

$f=(1+\int+\int\int+\int\int\int+\dots)1$

$=1+\int_0^x1dx+\int_0^x\int_0^x1dx+\int_0^x\int_0^x\int_0^x1dx+\dots$

$=1+x+\frac{x^2}{2}+\frac{x^3}{6}+\dots$

=e^x

done…

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