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Today's calculation of Integral 796

Source: 2012 Yokohama National University entrance exam/Science, #4

February 27, 2012
calculusintegrationtrigonometrylogarithmsgeometrygeometric transformationrotation

Problem Statement

Answer the following questions:
(1) Let aa be non-zero constant. Find ∫x2cos⁡(aln⁡x)dx.\int x^2 \cos (a\ln x)dx.
(2) Find the volume of the solid generated by a rotation of the figures enclosed by the curve y=xcos⁡(ln⁡x)y=x\cos (\ln x), the xx-axis and the lines x=1, x=eπ4x=1,\ x=e^{\frac{\pi}{4}} about the xx-axis.
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