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Today's calculation of Integral 572

Source: 2010 Sapporo Medical University entrance exam

February 27, 2010
calculusintegrationtrigonometrylogarithmscalculus computations

Problem Statement

For integer n, an n,\ a_n is difined by a_n\equal{}\int_0^{\frac{\pi}{4}} (\cos x)^ndx. (1) Find a_{\minus{}2},\ a_{\minus{}1}. (2) Find the relation of an a_n and a_{n\minus{}2}. (3) Prove that a_{2n}\equal{}b_n\plus{}\pi c_n for some rational number bn, cn b_n,\ c_n, then find cn c_n for n<0 n<0.
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