# Definite integral: exponential with squared exponent

• jumble0469
In summary, the conversation discusses solving a Gaussian integral with the function f(x) = \int^\infty_{-\infty}ce^{yx-y^2/2} dy. The suggestion is made to complete the square and use the standard result for a Gaussian integral, \int_{-\infty}^\infty e^{-x^2 / 2} = \sqrt{2 \pi}. It is also noted that this integral is commonly used in theoretical physics.
jumble0469
Hi,

I'm trying to solve the following:
$$f(x) = \int^\infty_{-\infty}ce^{yx-y^2/2} dy$$
where c is a constant
My only idea thus far was that since it is an even function, the expression can be simplified to:
$$= 2c\int^\infty_0 e^{y(x-{1/2}y)} dy$$

but I'm stuck here.

Anyone know how to do this?

This is a Gaussian integral (which is ubiquitous in theoretical physics, so if you have any aspirations in that direction pay extra attention )

The trick is to complete the square: write*
$$y^2 - 2 y x = (y - a)^2 - b$$
where a and b are independent of y, so you get
$$f(x) = c e^{-b/2} \int_{-\infty}^{\infty} e^{- (y - a)^2 / 2} \, dy.$$
Then you can do a variable shift and use the standard result
$$I = \int_{-\infty}^\infty e^{-x^2 / 2} = \sqrt{2 \pi}$$
which you can easily prove (if you've never done it, try it: consider $I^2$ and switch to polar coordinates).

* From this line onwards, I take no responsibility for sign errors and wrong factors of 1/2 - please check yourself :)

## 1. What is a definite integral?

A definite integral is a mathematical concept that represents the area under a curve between two specific points on the x-axis. It is used to find the exact value of the area, rather than just an approximation.

## 2. What is an exponential function with a squared exponent?

An exponential function with a squared exponent is a mathematical function in the form of f(x) = a^x, where a is a constant and x is the independent variable. When the exponent is squared, the function will have a curved shape instead of a straight line.

## 3. How do you solve a definite integral with an exponential function with a squared exponent?

To solve a definite integral with an exponential function with a squared exponent, you can use integration techniques such as substitution or integration by parts. You can also use the fundamental theorem of calculus, which states that the definite integral of a function can be evaluated by finding its antiderivative and plugging in the upper and lower limits of integration.

## 4. What is the significance of a definite integral with an exponential function with a squared exponent?

A definite integral with an exponential function with a squared exponent is often used in mathematical models to represent population growth, radioactive decay, and other natural phenomena. It can also be used in engineering and physics to calculate work, distance, and other quantities.

## 5. Are there any real-life applications of a definite integral with an exponential function with a squared exponent?

Yes, there are many real-life applications of a definite integral with an exponential function with a squared exponent. Some examples include predicting the spread of diseases in epidemiology, calculating the growth of bacteria in microbiology, and modeling the stock market in finance.

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