We will begin with perhaps the most basic variation on the Gaussian integral:
Where a > 0.
While the method for solving this integral will be the exact same as the standard Gaussian integral, I wanted to cover this since I will be using the result for the other integrals.
All we have to do is use the same polar coordinates method as before, but make use of the u-substitution of u = -ar² instead of u = -r²:
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