Previous page : The integral of e^(-x^2) from –infinity to infinity
in a second way. Ok, could we possibly try a similar trick as with some of the other integrals and insert a constant that we then treat as a variable to differentiate? What if we try
This would give us
Perhaps not all that useful. If we try to take partial derivatives then we either need to find the primitive function of our original integrand, or we just get higher and higher powers of x. If we try the substitution u=x2 we end up dividing by square root of u instead of multiplying by x2.
A second try
Ok, maybe we should try to put our parameter somewhere else. To start with we have that
Next we put our put our parameter as our upper integration limit. We get
Next we use a similar trick as in the previous method. We square the integral.
Differentiate with respect to p
Next we take the derivative of this with respect to p– as with some of the previous integrals.
Next we do the substitution
This gives us
Undoing the differentiation
Next we need to undo the differentiation with respect to p, i.e. we need to find
To do this we do the substitution
Going back to J(p) we thus have
Finding the constant of integration
To find C we evaluate this for p=0.
But on the other hand we have that
Putting it all together
Now, finally we let p go toward infinity.
And we will thus get
I don’t remember where I saw this integral solved in this way the first time. I have reproduced it from memory and by reasoning, so it might not be done in completely the same way.Up a level : Integrals
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