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Calculus/Evaluate the integral with respect to a?

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Expert: - 9/23/2009


Question
Evaluate the integral from 0 to infinity (e^-ax)(sinx)/x dx with respect to a?

Thanks a Bunch!

Answer
Just an update on the last letter.
Note that as x < 1, to divide by x gives a greater value.
However, since sinx Taylor's series starts with x - x^3/6 + x^5/120 - ...,
the series for sinx/x  is 1 - x^2/6 + x^4/120 - ....

As x goes to zero, this clearly goes to 1 quite rapidly.

Since it is known that |sinx| <=1, multiplying by (sinx)/x clearly makes the integral smaller.

I pointed out that the integral was relative to -1/a.

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