I'm currently just starting integration of complex improper integrals, and I'm already struggling on the following integral:
$$I = \int\limits_{-\infty}^\infty \frac{(x+1)\sin(2x)}{x^2 + 2x + 2}dx = \int\limits_{-\infty}^\infty f(x)dx$$
So far I've thought about finding residues to apply Cauchy Residues Theorem after defining $I$ as $\lim\limits_{R\to\infty}\int\limits_{-R}^R f(x)dx$. However I'm not very comfortable with this integral, and I'm not sure my method can be applied.. could anyone tell me if that's a right way to start, or give me a clue if not?
That would be great!