MAT334-2018F > End of Semester Bonus--sample problem for FE

FE Sample--Problem 6

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Victor Ivrii:
Calculate
$$\int_0^\infty \frac{x\sin (x)}{1+x^4}.$$

Hint:
Consider
$$\int _\Gamma f(z)\,dz \qquad \text{with } \ f(z)=\frac{ze^{iz}}{1+z^4}$$
over contour $\Gamma$ on the picture below:

Wanying Zhang:
Here is the solution of problem 6.

Wanying Zhang:
Sorry, I think I had a little computation mistake in the last photo.
Here is the new solution. Sorry for the inconvenience.

Victor Ivrii:
Contour integral is calculated correctly, but then with integral
$$\int_{-\infty}^\infty \frac{xe^{ix}}{1+x^4}dx$$
you are wrong: $e^{-ix}$ appears only after you transform it
$$\int_{-\infty}^\infty \frac{xe^{ix}}{1+x^4}dx= \int_0^\infty \frac{xe^{ix}}{1+x^4}dx +\int_{-\infty}^0 \frac{xe^{ix}}{1+x^4}dx = \int_0^\infty \frac{xe^{ix}}{1+x^4}dx +\int_{\infty}^0 \frac{-xe^{-ix}}{1+x^4}dx = \int_0^\infty \frac{x(e^{ix}-e^{-ix})}{1+x^4}dx = 2i I$$ .