Mathematics

Mathematics

Problem

Which of the following properties of the definite integral is ALWAYS true for any continuous function ff on the interval [a,b][a,b]? a) ∫abf(x) dx=∫baf(x) dx\displaystyle \int_a^b f(x)\,dx = \int_b^a f(x)\,dx b) ∫abf(x) dx=f ⁣(a+b2)(b−a)\displaystyle \int_a^b f(x)\,dx = f\!\left(\tfrac{a+b}{2}\right)(b-a) c) ∫abf(x) dx=∫acf(x) dx+∫cbf(x) dxfor every c between a and b\displaystyle \int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx\qquad\text{for every }c\text{ between }a\text{ and }b d) ∫abf(x) dx=∫acf(x) dx−∫cbf(x) dxfor every c between a and b\displaystyle \int_a^b f(x)\,dx = \int_a^c f(x)\,dx - \int_c^b f(x)\,dx\qquad\text{for every }c\text{ between }a\text{ and }b

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