# Integrate X²e^-x²: Solving a Tricky Integral

• arsmath
In summary, the person is trying to integrate the function e^-x² on the interval from -∞ to ∞, but they are having trouble. They have found a proof that e^-x² equals √∏/2 on this interval, but when they try to integrate by parts, they get an error. They are asking for help from the other people in the conversation, but they are not sure if they are supposed to learn this integration method in a Calculus course or if they can find help elsewhere.
arsmath
I am working on an integral I am finding tricky, and I think I'm missing something.
I need to integrate on the interval 0 to infinity, x²e^-x².
We have proved that on the interval of -∞ to ∞, e^-x²=√∏ so from o to ∞, it equals √∏/2. I can use this in my proof, but I don't see how. When I try integrating by parts I have trouble getting a finite answer. I would love some help,

Try integrating by parts the

$$I=\int_0^\infty e^{-x^2}\,d\,x$$

Rainbow Child said:
Try integrating by parts the

$$I=\int_0^\infty e^{-x^2}\,d\,x$$
HOw can one integrate this by parts, i do not think this has any closed form does it?

and this question is already on another forum!lol

HOw can one integrate this by parts, i do not think this has any closed form does it?

Like this:

$$I=\int_0^\infty e^{-x^2}\,d\,x=\int_0^\infty (x)'\,e^{-x^2}\,d\,x$$

and

$$I=\frac{\sqrt{\pi}}{2}$$

by OP

Rainbow Child said:
Like this:

$$I=\int_0^\infty e^{-x^2}\,d\,x=\int_0^\infty (x)'\,e^{-x^2}\,d\,x$$

and

$$I=\frac{\sqrt{\pi}}{2}$$

by OP

At what level is one supposed to learn how to integrate this?? I mean where is it covered?
Becasue this is my first time seeing such a trick!

Did you read the original post?
They gave him the result $I=\frac{\sqrt{\pi}}{2}$.

As for the actual calculation, there are many ways to calulate $I$. The simplest one is by double integrals.

my fault

sutupidmath said:
and this question is already on another forum!lol

i posted in the other forum before finding this one which I think may be more appropriate.

Rainbow Child said:
Did you read the original post?
They gave him the result $I=\frac{\sqrt{\pi}}{2}$.

As for the actual calculation, there are many ways to calulate $I$. The simplest one is by double integrals.

Well, i did not read the op's post!
and as for evaluating that integral, i think i should wait a few more months.

arsmath said:
i posted in the other forum before finding this one which I think may be more appropriate.
ok then, like Hurky said, show what u did so far?

sutupidmath said:
At what level is one supposed to learn how to integrate this?? I mean where is it covered?
Becasue this is my first time seeing such a trick!
I'm studying improper integrals for a "topics in advanced math" course. . .the proof is lengthy and involves double integrals and polar coords

arsmath said:
I'm studying improper integrals for a "topics in advanced math" course. . .the proof is lengthy and involves double integrals and polar coords
AH, i have never worked with double integrals, so i guess i cannot be of any further help, but surely the other guys will give u enough hints to get it right!
good luck!

## What is the definition of an integral?

An integral is a mathematical concept that represents the area under a curve in a graph. It is used to calculate the total value of a function over a specific interval.

## What is the general formula for solving integrals?

The general formula for solving integrals is ∫f(x) dx = F(x) + C, where f(x) is the function to be integrated, F(x) is the antiderivative (or indefinite integral) of f(x), and C is the constant of integration.

## How do you solve a tricky integral like X²e^-x²?

To solve this integral, you can use the substitution method where you substitute u = x² and du = 2x dx. This will transform the integral into ∫ue^-u du, which can be solved using integration by parts.

## What is the purpose of solving integrals in science?

Solving integrals in science allows us to calculate important quantities such as displacement, velocity, acceleration, and work. It is also used in various fields of science such as physics, chemistry, and engineering to model and analyze real-world phenomena.

## Are there any tips for solving tricky integrals more efficiently?

One tip for solving tricky integrals is to practice and become familiar with different integration techniques such as substitution, integration by parts, and trigonometric substitutions. It is also helpful to break down the integral into smaller parts and use algebraic manipulation to simplify the expression before integrating.

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