GATE 2025 ES – Question 49
Evaluate $\int_0^\infty\sin(4x)/(\pi x)\,dx$,to two decimals.
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Show answer and explanation
Correct answer: 0.50 to 0.50
Explanation
For $a>0$,the Dirichlet integral is $\int_0^\infty\sin(ax)/x\,dx=\pi/2$. One derivation inserts damping $e^{-\epsilon x}$,integrates to $\arctan(a/\epsilon)$,then takes $\epsilon\downarrow0$. With a=4 and the extra factor $1/\pi$,the result is **0.50**. This is an improper, conditionally convergent integral.