# Two Answers to Same Integral: Which One is Correct?

• siresmith
In summary: Integrating by parts gives the wrong answer because the boundary term, which is 1/2B, comes back into the integral and messes it up.
siresmith

## Homework Statement

Ive been giong mental over this. There seems to be two different answers to the same integral. I can't work out which is correct.

Integral from 0 to infinity of:

x(squared) exp(-2Bx(squared))

## The Attempt at a Solution

Integration by parts gives the answer as: -1/2B

My course lecturer gives the answer as: (1/8B) x route of (pie/2B)

I think she must have used the general formula:

Integral from -infinity to + infinity of x(power2n)exp(-Ax(squared)) = [1x3x5...(2n-1)]x(pie/A)/2(powern)A(powern)

and divided it by 2 as we only want the integral from 0 to infinity.

Whos right?? Can anyone work this out?
Please! write it down and see if you get the answer i did by integration by parts. If you do, do you think the lecturer is wrong?

Ok, used the integrator and... It came out as the standard integral suggests.

But why ia the integration by parts method wrong?? I am sure i did it right. does integration by parts not work on certain functions or something weird like that? Or did i just get it wrong- what do you get?

siresmith said:
But why ia the integration by parts method wrong?? I am sure i did it right. does integration by parts not work on certain functions or something weird like that? Or did i just get it wrong- what do you get?
If you show us your working, perhaps we could point out where you've gone wrong.

Hi siresmith!

If you integrate $$\int x^2 e^{-2Bx^2}$$ by parts,

you still end up with $$\int e^{-2Bx^2}$$ … didn't you?

Feynman trick-- take the derivative of I with respect to B.

$$I= \int e^{-2Bx^2}$$

If you know that integral already you will already be done. If not integrate your integral $$\frac{dI}{dB}$$ by parts and to find it as an expression in terms of I. Solve the de by separating and integrating.

But you should already know what I is, it's a classic result and it must have been given to you.

I am right that in integration by parts you let u= x and $dv= xe^{-Bx^2}$?

Yup and the boundary term vanishes, the odd integral vanishes and you're left with something like I/2B.

## What is an "integral"?

An integral is a mathematical concept that represents the area under a curve in a graph. It is used to solve problems involving rates of change, accumulation, and many other applications.

## What does it mean to have "2 answers" to the same integral?

In some cases, when evaluating an integral, there may be two different solutions that are both valid. This can happen when the function being integrated has a discontinuity or when the limits of integration are not clearly defined.

## How do you find the two answers to the same integral?

To find the two answers to the same integral, you can use different techniques such as substitution, integration by parts, or partial fractions. It is important to carefully analyze the problem and choose the appropriate method to find both solutions.

## Why is it important to find both answers to the same integral?

Having two solutions to the same integral can provide a deeper understanding of the problem and can help to verify the accuracy of the solutions. It also allows for more flexibility in solving mathematical equations.

## Are there any real-life applications of finding two answers to the same integral?

Yes, there are many real-life applications where finding two answers to the same integral is useful. For example, in physics, when calculating the work done by a variable force, there may be two different solutions depending on the starting point of the object. In finance, when calculating the present value of an investment, there may be two different solutions depending on the rate of return.

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