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$$\int_0^2f(x)\,\mathrm dx=\int_0^2 f(t)\,\mathrm dt.$$

I would say true because a&b = a&b and it would not make a change if is x,d,f,q,w,t it will logically be all the same, am I right ? but because it is MATH not logic so everything could be unexpected!

Hakim
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John
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1 Answers1

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It is of course true, because in a definite integral, the variable of integration is often just considered a "dummy variable".

Let's say the indefinite integral is $F(x) + c_1$ in one case, and obviously, $F(t) + c_2$ in the other, where $c_1$ and $c_2$ are arbitrary constants.

In both cases, the definite integral would work out to be $F(2) - F(0)$. The constants always vanish in a definite integral so they are irrelevant.

Deepak
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  • Nice explanation. Upshot is that, regardless of the variable name, you end up substituting definite values for it, erasing any "memory" of the variable used to perform the integration. Good, +1. – MPW Jul 09 '14 at 01:46