# Is the Integral of an Odd Function with a Non-Centered Even Exponential Zero?

• atlantic
In summary, the given conversation discusses the integral \int^{\infty}_{-\infty}ie^{-(x-x_{0})^{2}}sin(vx) dx, where x_0 and v are real constants. The experts suggest using the fact that the sine function is odd, and consider the behavior of the function as x approaches positive and negative values. They also mention using the trigonometric identity sin(vx) = sin(v(x-x_0))cos(vx_0)+cos(v(x-x_0))sin(vx_0) to simplify the integral.
atlantic

## Homework Statement

I have the integral:
$\int^{\infty}_{-\infty}ie^{-(x-x_{0})^{2}}sin(vx) dx$

where x_0 and v are real constants.

The sine function is odd. But what about the exponential? If it's even, then the integral is zero, but the exponential is not centred around origo. Will the integral still be zero as long as the exponential is not an odd function?

hi atlantic!

no

try using sin(vx) = sin(v(x - xo))cos(vxo) + cos(v(x - xo))sin(vxo)

The first thing I would do is to let $x=\pm a$ and compare the two, what would you expect to happen if the function was odd?

## What is an even function?

An even function is a mathematical function where f(-x) = f(x) for all values of x. In other words, when the input of the function is replaced by its negative equivalent, the output remains the same. Graphically, an even function is symmetrical about the y-axis.

## What is an odd function?

An odd function is a mathematical function where f(-x) = -f(x) for all values of x. This means that when the input of the function is replaced by its negative equivalent, the output is the negative of the original output. Graphically, an odd function is symmetrical about the origin.

## What is the integral of an even function?

The integral of an even function from -a to a is equal to twice the integral of the function from 0 to a. In other words, the integral of an even function over a symmetric interval is equal to the area under the curve on one half of the interval multiplied by 2.

## What is the integral of an odd function?

The integral of an odd function over a symmetric interval is equal to 0. This is because the positive and negative areas under the curve cancel each other out, resulting in a net area of 0.

## Can an even/odd function have a non-zero integral?

Yes, an even/odd function can have a non-zero integral if the interval of integration is not symmetric. In this case, the areas under the curve on either side of the y-axis do not cancel out, resulting in a non-zero integral.

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