Toronto Math Forum
MAT3342018F => MAT334Tests => End of Semester Bonussample problem for FE => Topic started by: Victor Ivrii on November 27, 2018, 03:57:44 AM

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:

Here is the solution of problem 6.

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

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
$$\int_{\infty}^\infty \frac{xe^{ix}}{1+x^4}dx= \int_0^\infty \frac{xe^{ix}}{1+x^4}dx +\int_{\infty}^0 \frac{xe^{ix}}{1+x^4}dx = \int_0^\infty \frac{xe^{ix}}{1+x^4}dx +\int_{\infty}^0 \frac{xe^{ix}}{1+x^4}dx = \int_0^\infty \frac{x(e^{ix}e^{ix})}{1+x^4}dx = 2i I$$ .