From the First Fundamental Theorem, we had that `F(x) = int_a^xf(t)dt` and `F'(x) = f(x)`.
Suppose `G(x)` is any antiderivative of `f(x)`. (Remember, a function can have an infinite number of antiderivatives which just differ by some constant, so we could write `G(x) = F(x) + K`.)
So we'll have:
`G'(x) = F'(x)`
Now, since `G(x) = F(x) + K`, we can write:
`G(b) - G(a) = (F(b)+K) - (F(a) + K)`
`=F(b) - F(a)`
`=int_a^bf(t)dt - int_a^af(t)dt`
`=int_a^bf(t)dt - 0`
So we've proved that `int_a^bf(x)dx = F(b) - F(a)`.
Note: Once again, when integrating, it doesn't really make any difference what variable we use, so it's OK to use `t` or `x` interchangeably, as long as we are consistent.
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