# Integrate x/(x^2-2x+5): Clarifying Steps

• Azureflames
In summary, The conversation is about finding the integral of x/(x^2-2x+5) dx and the steps involved in solving it. The individual started by completing the square in the denominator and then substituted back in to get (1/2)ln(x^2-2x+5) - (1/2)arctan((x-1)/2) + C as the final answer. They also mention using partial fraction decomposition as an alternative method.
Azureflames

## Homework Statement

Hi, I'm trying to take the integral of x/(x^2-2x+5) dx but I'm not sure what to do.

## The Attempt at a Solution

I started by completeing the square in the denominator giving me the integral of x/((x-1)^2+4)) dx but I am not sure where to go from there. I have the correct answer for it but I need to understand the steps involved getting there.

EDIT: Okay, after spending a decent amount of time on this problem I finally look for some place to get help, and 5 minutes later I think I come up with a solution.

First, I set u = x^2 - 2x +5, du/2 - 2/2 = x. Substituted back in which gave me (1/2)integral( (du-2)/u ) which I then split into (1/2)integral(du/u) - (1/2)integral(2/u).

Taking the integral of the first part gave me (1/2)ln(x^2 - 2x +5). For the second half, I substitued the u values back into the equation which gave me: -(1/2)integral(2/(x^2 - 2x + 5). I completed the square in the denominator which gave me -integral( 1/((x-1)^2+4 ).

Integrating that part of the equation gives me -(1/2)arctan((x-1)/2).

So my final answer is (1/2)ln(x^2 - 2x +5) - (1/2)arctan((x-1)/2) + C. Can someone please confirm my steps? Sorry if my work is hard to follow. I wasn't sure how to make the proper symbols and my time is short :)

Last edited:
That's right.

Hell, that seemed clever(or I suck)

I would've done partial fraction decomposition, and heaven knows NO ONE likes doing that

## 1. What is the purpose of integrating x/(x^2-2x+5)?

The purpose of integrating x/(x^2-2x+5) is to find the antiderivative or the indefinite integral of the given function. This helps in solving problems related to motion, area, volume, and other real-life applications.

## 2. What are the steps to integrate x/(x^2-2x+5)?

The steps to integrate x/(x^2-2x+5) are as follows:

1. Complete the square to rewrite the denominator as (x-1)^2+4.
2. Use the substitution u = x-1 to simplify the integral to ∫du/(u^2+4).
3. Apply the formula for the inverse tangent function to evaluate the integral as (1/2)tan^-1(u/2) + C.
4. Substitute back u = x-1 to get the final result as (1/2)tan^-1((x-1)/2) + C.

## 3. Why do we need to complete the square in the first step?

Completing the square allows us to rewrite the denominator in a form that is easier to integrate. This helps in simplifying the integral and making it possible to apply various integration techniques.

## 4. Can we solve the integral without using substitution?

Yes, the integral x/(x^2-2x+5) can also be solved using the method of partial fractions. However, the process can be more complex and time-consuming compared to using the substitution method.

## 5. Is there a way to check if the integration is done correctly?

Yes, one can differentiate the antiderivative obtained to see if it results in the original function x/(x^2-2x+5). If the answer is yes, then the integration has been done correctly.

• Calculus and Beyond Homework Help
Replies
15
Views
859
• Calculus and Beyond Homework Help
Replies
10
Views
622
• Calculus and Beyond Homework Help
Replies
10
Views
1K
• Calculus and Beyond Homework Help
Replies
22
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
570
• Calculus and Beyond Homework Help
Replies
14
Views
531
• Calculus and Beyond Homework Help
Replies
27
Views
2K
• Calculus and Beyond Homework Help
Replies
12
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
832