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i1) Evaluate\( \int_{0}^{2 \pi} \sin ^{4} x \cos ^{6} x d x \) |
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Answer» \(\int\limits_0^{2\pi}\)sin4x cos6x dx = 4\(\int\limits_0^{\pi/2}\)sin4x cos6x dx \(=\cfrac{4\Gamma(\frac{4+1}2)\Gamma(\frac{6+1}2)}{2\Gamma(\frac{4+6+2}2)}\) \(=\cfrac{2\Gamma(\frac52)\Gamma(\frac72)}{\Gamma(6)}\) \(=\frac{2\times3/2\times1/2\,\Gamma(1/2)\times5/2\times3/2\times1/2\,\Gamma(1/2)}{(5\times4\times3\times2\times1)}\) \(=\frac{3}{16\times8}\pi.\pi\) (\(\because\) \(\Gamma(1/2)=\sqrt{\pi}\)) \(=\frac{3\pi}{128}\) |
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