Exercise · ZW-REAL-INTEGRAL-BY-RESIDUES-005 · ★☆☆ Basic
Compute \(\int_{-\infty}^{\infty}\frac{dx}{x^{2}+2x+2}\).
Choose one option.
Show answer
- C. \(\pi\)
Explanation
Since \(x^2+2x+2=(x+1)^2+1\), the poles are \(-1\pm i\). The upper one is \(-1+i\), with residue \(1/(2i)\), and the integral is \(\pi\).
Review the concept: Real integral by residues
A method to compute \(\int_{-\infty}^{\infty} f(x)\,dx\) with \(f\) rational: close the real axis with a semicircle in the upper half-plane and apply the residue theorem.
\[\int_{-\infty}^{\infty} f(x)\,dx=2\pi i\sum_{\operatorname{Im}z_k>0}\operatorname{Res}(f,z_k)\]
More practice problems: real integral by residues
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- ZW-REAL-INTEGRAL-BY-RESIDUES-008 Compute ∫_-∞^∞(dx)/(x^4+1).
- ZW-REAL-INTEGRAL-BY-RESIDUES-009 Let f=P/Q be rational with no real poles. Which condition guarantees that the integral over the sem…
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