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Integrate from 0 to Pi ecos t sin 2t dt?
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Your typography is ambiguous. Do you mean e^(cos t) * sin(2t) dt? I would let w = cos(t), then dw = -sin(t) dt, and the integrand is now -we^w dw. The best way to deal with this is to integrate by parts; it comes out to -we^w+e^w+C. Now go back to "t," and you have -cos(t)e^(cos t) + e^(cos t). When t = pi, this is 1/e + 1/e; when t = 0, it's -e + e = 0. So apparently the value is 2/e, but you'd better check my details.