# Integrating Rational Functions with Substitution

• binbagsss
In summary, the conversation discusses the attempted solution to the given integral, which involves using integration by parts and substitution. However, there is a discrepancy between the solution obtained and the expected solution. The conversation then suggests using a different approach, such as substitution, to solve the integral.
binbagsss

## Homework Statement

Apologies if this is obvious, maybe I'm a little out of touch

## \int\limits^b_0 \frac{x^3}{x^2+m^2} dx ##

## The Attempt at a Solution

I [/B]was going to go by parts breaking the ##x^3 = x^2 . x##

So that I have the logarithm

I.e :

##b^2 \frac{log (b^2 + m^2)}{2} - \int \frac{log (x^2+m^2)}{2} 2x dx ##
But the solution is :

## b^2 + m^2 log ( \frac{m^2}{b^2+m^2} ) ##( I thought that perhaps the solution could be going by parts again, but there is no reason for the boundary term to vanish )

Ta

binbagsss said:

## Homework Statement

Apologies if this is obvious, maybe I'm a little out of touch

## \int\limits^b_0 \frac{x^3}{x^2+m^2} dx ##

## The Attempt at a Solution

I [/B]was going to go by parts breaking the ##x^3 = x^2 . x##

So that I have the logarithm

I.e :

##b^2 \frac{log (b^2 + m^2)}{2} - \int \frac{log (x^2+m^2)}{2} 2x dx ##
But the solution is :

## b^2 + m^2 log ( \frac{m^2}{b^2+m^2} ) ##( I thought that perhaps the solution could be going by parts again, but there is no reason for the boundary term to vanish )

Ta
The second term in your answer can be integrated using substitution, with ##u = x^2 + b^2, du = 2xdx##.

Delta2
Write ##x^3=x(x^2+m^2)-m^2x## so the integral becomes ##\int xdx-\int \frac{m^2x}{x^2+m^2}dx##.

Or just use the substitution ##u=x^2+m^2## on the original integral.

## What is quick integration?

Quick integration is a method used by scientists to combine different pieces of information or data into a cohesive understanding or model. It allows for efficient and effective analysis of complex systems or phenomena.

## Why is quick integration important in the scientific process?

Quick integration is important because it allows scientists to make connections between various pieces of information and data, leading to a deeper understanding of a particular subject or research question. It also helps to identify patterns and relationships that may not be obvious when looking at individual pieces of information.

## What are some common challenges when using quick integration?

Some common challenges when using quick integration include dealing with large amounts of data, determining which pieces of information are relevant and how to combine them, and addressing any inconsistencies or uncertainties in the data.

## What are some strategies for effectively using quick integration?

To effectively use quick integration, scientists should have a clear understanding of the research question or problem they are trying to address, carefully select and evaluate the data they will be integrating, and use appropriate analytical techniques and tools to combine the data and draw meaningful conclusions.

## Can quick integration be applied to all fields of science?

Yes, quick integration can be applied to all fields of science. While the specific methods and techniques may vary depending on the discipline, the overall goal of integrating different pieces of information to gain a deeper understanding remains the same.

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