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

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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.
  • #1
Azureflames
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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.

Homework Equations





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 :)
 
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  • #2
That's right.
 
  • #3
Hell, that seemed clever(or I suck)

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

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

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.

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