# How to compute the following integral

• thenewbosco
In summary, the conversation was about how to compute a specific integral that involved a term in the exponent with x times x0. The suggested method was to factor the expression and make a substitution, but there was uncertainty about the correct substitution to use. It was also mentioned that integration by parts may not work for this integral.
thenewbosco
I was wondering how to compute the following integral:
$$\int xe^{\frac{-x^2+2x\cdot x_0-x_{0}^2}{2a^2}}dx$$

it would be quite simple if there was no term in the exponent x times x0 i think...any help on this thanks

Factor the expression in the exponent and then make a substitution.

i factored the exponent to be -(x-x0)^2 but what is the substitution to make? u = x-xo? this does not help because of the x...

use integration by parts after making the substitution.

so the substitution i proposed is the correct one to make?

Try u=(x-x0)2, I think that should work, but it might be a bit messy, I'm pretty sure parts will not work though.

but for what you propose then du=2(x-xo)dx, which cannot be put into the integral like this...

It has to have some error function in it, so definitely it's not an elementary integral.

Daniel.

## 1. What is an integral?

An integral is a mathematical concept that represents the area under a curve on a graph. It is used to find the total value of a function over a given interval.

## 2. How do you compute an integral?

To compute an integral, you must first find the antiderivative of the function. This is done by using integration techniques, such as substitution, integration by parts, or partial fractions. Once the antiderivative is found, you can evaluate the integral by plugging in the upper and lower limits of the given interval.

## 3. What is the difference between a definite and indefinite integral?

A definite integral has specific limits of integration and gives a numerical value as a result. An indefinite integral does not have limits and gives an expression with a constant as a result.

## 4. What are the different types of integrals?

The main types of integrals are definite and indefinite integrals. Other types include improper integrals, line integrals, surface integrals, and volume integrals. These types are used to solve different types of problems in various fields of mathematics and science.

## 5. How is integration used in real life?

Integration is used in many real-life applications, such as calculating areas and volumes in engineering, finding the center of mass in physics, and determining probabilities in statistics. It is also used in economics, biology, and other fields to model and solve problems involving continuous data.

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