Exercise · ZW-REAL-INTEGRAL-BY-RESIDUES-010 · ★★★ Advanced
Compute \(\int_{-\infty}^{\infty}\frac{dx}{(x^{2}+1)^{3}}\).
Choose one option.
Show answer
- C. \(3\pi/8\)
Explanation
The pole \(i\) is triple; its residue is \(\tfrac12\,\tfrac{d^2}{dz^2}(z+i)^{-3}\) evaluated at \(i\), that is \(\tfrac{3}{16i}\). Then \(2\pi i\cdot\tfrac{3}{16i}=3\pi/8\).
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
- ZW-REAL-INTEGRAL-BY-RESIDUES-001 Compute ∫_-∞^∞(dx)/(x^2+1).
- ZW-REAL-INTEGRAL-BY-RESIDUES-002 Compute ∫_-∞^∞(dx)/(x^2+4).
- ZW-REAL-INTEGRAL-BY-RESIDUES-003 For ∫_-∞^∞(dx)/(x^2+9), which poles of 1/(z^2+9) enter the sum of residues?
- ZW-REAL-INTEGRAL-BY-RESIDUES-004 Compute Res\!((1)/(z^2+1), i).
- ZW-REAL-INTEGRAL-BY-RESIDUES-005 Compute ∫_-∞^∞(dx)/(x^2+2x+2).
- ZW-REAL-INTEGRAL-BY-RESIDUES-006 Compute ∫_-∞^∞(dx)/((x^2+1)^2).
- ZW-REAL-INTEGRAL-BY-RESIDUES-007 Compute ∫_-∞^∞(dx)/((x^2+1)(x^2+4)).
- ZW-REAL-INTEGRAL-BY-RESIDUES-008 Compute ∫_-∞^∞(dx)/(x^4+1).