What is the product of the integration of e^(0.5x^2)?

Question

What is the product of the integration of e^(0.5x^2)? I tried integration by parts but I can't get it so can anyone show me the steps and whether we use integration by parts or normal integration?

This integration must be done numerically. There is no known closed form of the integral.
How do you know that? and how do you do the integration numerically with the lower limit 0 and upper limit 2?

Wow, you ppl are so good

Answer 1

PSV is correct and gives you a good cite. HOWEVER the trick that is used to find the integral only works on integrals form -infintiy to + infinity. If you want to find this integral from zero to 2, you need to do it by creating a series and integrating term by term.

e^x = 1 + x + x^2/2! + x^3/3! + x^4/4! + x^5/5! + ....

e^(x^2/2) = 1 + (x^2/2) + x^4/(2^2(2!)) + x^6/(2^3(3!) + ...

Then you integrate term by term

Int[e^(x^2/2)] = x + x^3/(3*2) + x^5/(5*2(2!)) + x^7/(7*2^3(3!)) + ...

Finally, you evaluate at 0 and 2. Evaluation at 0 gives 0.

Evaluation at 2 gives the final answer which is

2 + 8/6 + 32/20 + 128/252 + ....

Answer 2

This integration must be done numerically. There is no known closed form of the integral.

Answer 3

This integral is what you call the Gaussian integral and can't be solved by normal methods but by using a trick. Check Mathworld to see the solution. b-)