# Evaluate the integral as infinit series.

• rcmango
In summary, an integral is a mathematical concept used to find the accumulation of a quantity over an interval. An infinite series is a sum of an infinite number of terms and is often used in calculus to approximate values of functions. To evaluate an integral as an infinite series, the Maclaurin series expansion technique is used. This can be applied to solve difficult integrals and has applications in physics and engineering. However, there are limitations to this technique, as not all integrals can be evaluated as infinite series and the accuracy of the results may vary.
rcmango

## Homework Statement

Evaluate the indefinite integral as an infinite series.

here is the problem: http://aycu28.webshots.com/image/38987/2005883203189533192_rs.jpg

## The Attempt at a Solution

not sure what test to use. My first guess would be to use the integral test from 1 to infinity? i know derivative of e^x is e^x. I may have to do this by parts.

Thanks.

You could expand the inside of the integral into a Taylor/McLaurin series, and then integrate that term-by-term.

## 1. What is an integral?

An integral is a mathematical concept that represents the accumulation of a quantity over a given interval. It is used to find the area under a curve, the volume of a solid, and many other applications in mathematics and science.

## 2. What is an infinite series?

An infinite series is a sum of an infinite number of terms. It is often written in the form of sigma notation, where the terms are added together using a specific pattern or formula. Infinite series are used in calculus to approximate values of functions and solve various problems.

## 3. How do you evaluate an integral as an infinite series?

To evaluate an integral as an infinite series, you need to use a technique called Maclaurin series expansion. This involves using a known series or formula to rewrite the integrand (the expression being integrated) in a form that can be more easily integrated. Then, you can apply the power rule for integration to find the infinite series representation of the integral.

## 4. What are the applications of evaluating an integral as an infinite series?

One of the main applications of evaluating an integral as an infinite series is to solve difficult or impossible integrals. By using Maclaurin series expansion, we can convert the integral into a series that can be more easily integrated. This technique is also used in physics and engineering to approximate solutions to various problems.

## 5. Are there any limitations to evaluating an integral as an infinite series?

Yes, there are limitations to this technique. Not all integrals can be evaluated as infinite series, and some may require advanced mathematical knowledge to convert into a series form. Additionally, the accuracy of the series representation depends on the number of terms used, so it may not always provide an exact solution. It is important to use caution and double check the results when using this method to evaluate integrals.

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