improper integral

Question

http://www.questionpapers.net.in/Question%20Papers/gate-question-papers-download/gate-question-papers-download-civil-engineering-2007.pdf

question 25....plz give it proper solution

Comments

Is improper integral in Gate CS syllabus?

Answer 1

I think it could be done by laplace transformation

http://www.freemathhelp.com/forum/threads/56392-Integral-using-Laplace-Transform-int-0-infty-sin-t-t-dt

Answer 2

$\bf{\displaystyle I(a,b) = \int_{0}^{\infty}\frac{e^{-ax}\sin (bx)}{x}dx = \int_{0}^{\infty}\int_{0}^{b}e^{-ax}\cos(yx)dydx}$

$\bf{\displaystyle I(a,b) = \int_{0}^{b}\int_{0}^{\infty}e^{-ax}\cos(yx)dxdy=\int_{0}^{\infty}\left[\frac{e^{-ax}}{a^2+y^2}\left(y\sin (yx)-acos (yx)\right)\right]_{0}^{\infty}dy}$

$\bf{\displaystyle I(a,b) = \int_{0}^{b}\frac{a}{y^2+a^2}dy = \tan^{-1}\left(\frac{b}{a}\right)}$

$\bf{\displaystyle I(0,1) =\int_{0}^{\infty}\frac{\sin x}{x}dx =  \tan^{-1}\left(\frac{1}{0}\right) = \tan^{-1}(\infty) = \frac{\pi}{2}}$