29 November 2020 3 8K Report

I came across this question (attached below). I tried to solve it but got stuck in the portion where we need to calculate the inverse tranform. I found a solution of this question (also attached below) but there, in the encircled portion, I couldn't get how they took a 2x factor out and change the limits from (-∞, +∞) to [0,+∞ ). I know that we can do this changing of limits only if it's an even function and can take the limiting points [0,+∞ ) iff the function is of the form y = ax2 . But here the term inside the exponential function is of the form y = ax2 + bx + cix, where i =complex number, and accordingly the limit should change to some random [m,+∞ ) in place of [0,+∞ ). Also, the 2x factor would not be there because the limiting point is changing from [0,+∞ ) to [m,+∞ ) and the graph will not be symmetrical across X=0 Axis.

I will be highly grateful if you can kindly clarify my doubt or let me know where I am making a mistake in understanding the question.

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