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I am having difficulty integrating $\int (y + \sin(e^{x^2})) \, dx - 2x \, dy$, over the circle $x^2 + y^2 = 1$, traversed anti-clockwise.

I managed to do the differentiation and convert everything in terms of t, but I reach this step in the integration process, $-3\pi - \int_0^{2\pi} \sin(e^{\cos^2 t}) \sin(t) \, dt$, and then I don't know how to proceed from here. I tried integrating by parts but it doesn't seem to work.

Does anyone know how I should solve this problem? Thanks so much!

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  • $\begingroup$ That's an odd function you're trying to integrate. $\endgroup$ – Randy E Jan 10 '15 at 23:47
  • $\begingroup$ The function is also periodic with period $2\pi,$ so you can integrate over any interval of length $2\pi,$ perhaps one that is symmetric about 0. (In other words, you're still integrating over the circle, with the same parametrization, but with a different interval for $t$). $\endgroup$ – Randy E Jan 11 '15 at 0:19
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Using Green's Theorem, $\int_{C}(M dx + N dy)=\iint_{R}(N_{x}-M_{y}) \;dA=\iint_R(-3) \;dA=-3A=-3\pi.$

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  • $\begingroup$ Thanks! May I know how should we know to use Green's Theorem in this case? Is it a method of last resort in such situations if integration does not work (and assuming conditions are satisfied)? $\endgroup$ – thbcm Jan 11 '15 at 6:07
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    $\begingroup$ If the line integral isn't easy to calculate directly (in particular, when the curve is composed of pieces that must be parametrized separately), then Green's Theorem is a good alternative to consider. $\endgroup$ – user84413 Jan 11 '15 at 22:09

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