For real numbers and an integrable function

MATHEMATICS · Single correct · AP · Calculus AB · Integration and Accumulation of Change · Integration and Accumulation of Change

Problem

For real numbers a<b<ca<b<c and an integrable function gg, which of the following identities holds for all such gg? (a) ∫abg(x) dx=∫bag(x) dx\displaystyle \int_{a}^{b} g(x)\,dx = \int_{b}^{a} g(x)\,dx (b) ∫acg(x) dx=−∫cag(x) dx\displaystyle \int_{a}^{c} g(x)\,dx = -\int_{c}^{a} g(x)\,dx (c) ∫abg(x) dx+∫bcg(x) dx=−∫acg(x) dx\displaystyle \int_{a}^{b} g(x)\,dx + \int_{b}^{c} g(x)\,dx = -\int_{a}^{c} g(x)\,dx (d) ∫acg(x) dx=∫abg(x) dx+∫cbg(x) dx\displaystyle \int_{a}^{c} g(x)\,dx = \int_{a}^{b} g(x)\,dx + \int_{c}^{b} g(x)\,dx

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