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Derive an identity of integral formulas

For f an integrable function on [a,b] prove the following identity:

    \[ \int_a^b f(x) \, dx = (b-a) \int_0^1 f(a+(b-a)x) \, dx. \]


Proof. First, we use the theorem on invariance of the integral under translation (Theorem 1.18 of Apostol), to write,

    \[ \int_a^b f(x) \, dx = \int_0^{b-a} f(x+a) \, dx. \]

Then, using the expansion/contraction of the interval of integration (Theorem 1.19 of Apostol) we obtain

    \[ \int_0^{b-a} f(x+a) \, dx = (b-a) \int_0^1 f(x(b-a) + a) \, dx = (b-a) \int_0^1 f(a + (b-a)x). \qquad \blacksquare \]

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