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869

Part of 2013 Today's Calculation Of Integral

Problems(1)

Today's calculation of Integral 869

Source: 2013 Showa university entrance exam/Medicine, 2nd exam

3/15/2013
Let In=1n+1∫0πx(sin⁡nx+nπcos⁡nx)dx  (n=1, 2, ⋯ ).I_n=\frac{1}{n+1}\int_0^{\pi} x(\sin nx+n\pi\cos nx)dx\ \ (n=1,\ 2,\ \cdots).
Answer the questions below.
(1) Find In.I_n.
(2) Find ∑n=1∞In.\sum_{n=1}^{\infty} I_n.
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