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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Fall 2016 • Question 8

Question

Math 6420 • PDE Fall 2016 • Question 8

Integrate

  1. \[ \int _0^\infty \frac {\sin \alpha x}{x}\,dx, \]
  2. \[ \int _0^\infty \frac {x^\alpha }{1+x}\,dx, \]
  3. \[ \int _0^\infty \sin x^2\,dx \]

[Explain the logic (in particular, the choice of contour), but do not worry about getting the exact numbers; \(\alpha \) is a real parameter.]

Raw LaTeX

Integrate
\begin{enumerate}
\item[(a)]
\[
  \int_0^\infty \frac{\sin\alpha x}{x}\,dx,
\]

\item[(b)]
\[
  \int_0^\infty \frac{x^\alpha}{1+x}\,dx,
\]

\item[(c)]
\[
  \int_0^\infty \sin x^2\,dx
\]
\end{enumerate}
[Explain the logic (in particular, the choice of contour), but do not worry about getting
the exact numbers; \(\alpha\) is a real parameter.]

Topic Tags

Contour Integration and Residue Calculus Fourier Transform and Sampling Theory Linear Programming and Optimization Real Integral Evaluation by Complex Methods