Mathematics

Mathematics

Problem

For xx in [0,1][0,1], it is known that ex≥1+xe^{x}\ge 1+x. Which of the following correctly compares the two integrals ∫01ex dx\displaystyle\int_{0}^{1} e^{x}\,dx and ∫01(1+x) dx\displaystyle\int_{0}^{1} (1+x)\,dx? a) ∫01ex dx=∫01(1+x) dx\displaystyle\int_{0}^{1} e^{x}\,dx = \int_{0}^{1} (1+x)\,dx b) ∫01ex dx>∫01(1+x) dx\displaystyle\int_{0}^{1} e^{x}\,dx > \int_{0}^{1} (1+x)\,dx c) ∫01ex dx<∫01(1+x) dx\displaystyle\int_{0}^{1} e^{x}\,dx < \int_{0}^{1} (1+x)\,dx d) The relation cannot be determined without evaluating both integrals exactly

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