![]() The integral goes from 0 to infinity, and at x= infinity, sin( ax) and cos( ax) are not defined. We take the derivative with respect to a and we get xcos( ax)/ x=cos( ax) okay, this is an easy integral… well, no, and this is why: You may think we can do something like this: ![]() ![]() Generalized form of the integral of sin( x)/ x In this case, in order to make this integral easy, we need to get rid of that x at the denomitator to do so we need to differentiate a function that contains a parameter multiplied by x so that the x is generated and the one at the denominator will cancel out with it. The idea is that we introduce a parameter so that, when differentiating (with respect to the parameter) a certain function that contains it, something happens that will make the integral easier to evaluate. We need to use a different approach: differentiation under the integral sign (also known as Feynman technique). Today we have a tough integral: not only is this a special integral (the sine integral Si( x)), but it also goes from 0 to infinity! Because of the first characteristic, there is no elementary antiderivative and therefore we can’t simply plug the bounds into the result this means none of the techniques we know of will work. ![]()
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