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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6420 • PDE

Spring 2019 • Question 8

Question

Math 6420 • PDE Spring 2019 • Question 8

Evaluate the integrals \[ \begin {array}{rcll} \text {(a)}&\displaystyle \int _{-\infty }^{\infty }\frac {\sin ax}{x}\,dx &&\text {(\(a\) is a real parameter)},\\[2.2ex] \text {(b)}&\displaystyle \int _0^\infty \frac {x^\alpha }{x+1}\,dx &&\text {(\(\alpha \) is a real parameter, \(-1<\alpha <0\))}. \end {array} \]

Raw LaTeX

Evaluate the integrals
\[
\begin{array}{rcll}
\text{(a)}&\displaystyle \int_{-\infty}^{\infty}\frac{\sin ax}{x}\,dx
&&\text{(\(a\) is a real parameter)},\\[2.2ex]
\text{(b)}&\displaystyle \int_0^\infty\frac{x^\alpha}{x+1}\,dx
&&\text{(\(\alpha\) is a real parameter, \(-1<\alpha<0\))}.
\end{array}
\]

Topic Tags

Cauchy Integral Formula and Integral Theorems Fourier Transform and Sampling Theory Linear Programming and Optimization Real Integral Evaluation by Complex Methods