Previous page : The integral of e^(-x^2) from –infinity to infinity

The integral

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*=*x*^{2} we end up dividing by square root of *u* instead of multiplying by *x*^{2}.

**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

so

or

**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.

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