Word Count : 309

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The second mean value theorem for integrals states: if (f(x)) is monotonic on ([a,b]) and (g(x)) is integrable, then there exists a (c \in [a,b]) such that [ \int_a^b f(x) g(x) , dx = f(a) \int_a^c g(x) , dx + f(b) \int_c^b g(x) , dx. ] Here, (f(x) = x) (increasing, hence monotonic) and (g(x) = \cos x) on ([0, \pi/2]). First, calculate the left-hand side: [ \int_0^{\pi/2} x \cos x , dx. ] Use integration by ______ _________ __________ __________ ________ __________ _________ ________ _________.
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