How can Arctan and ln be used to simplify the integral of (x+1)/(x^2+4x+5)?

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Discussion Overview

The discussion revolves around the integration of the function (x+1)/(x^2+4x+5) using techniques involving the arctangent function and natural logarithm. Participants explore various strategies and approaches to simplify the integral, focusing on theoretical and mathematical reasoning.

Discussion Character

  • Mathematical reasoning, Homework-related, Technical explanation

Main Points Raised

  • One participant presents an approach to the integral by rewriting it as 1/2∫(2x+2)/(x^2+4x+5) dx and further decomposing it into two parts, suggesting a combination of logarithmic and arctangent functions for the solution.
  • Another participant emphasizes the importance of including the "constant of integration" in the final answer.
  • A different participant suggests re-writing the original expression to make it more manageable for integration.

Areas of Agreement / Disagreement

There is no clear consensus on the best approach to solve the integral, as participants present different strategies and emphasize various aspects of the solution process.

Contextual Notes

Some participants' suggestions depend on the manipulation of the original expression, which may involve assumptions about the forms of the functions used in integration.

TheTimeTraveler
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∫(x+1)/(x^2 + 4x +5) dx
Anyone can help me doing it using Arctan and ln
 
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Let's first hear out your own idea to solve this problem? What strategy you have in mind?
 
blue_leaf77 said:
Let's first hear out your own idea to solve this problem? What strategy you have in mind?
∫(x+1)/(x^2+4x+5) dx = 1/2∫ (2x+2)/(x^2+4x+5) dx = 1/2∫(2x+4)/(x^2+4x+5) dx + 1/2∫(-2)/(1+(x+2)^2) dx
[1/2 ln l(x^2+4x+5)l ] - arctan(x+2)
SO??
 
Don't forget the "constant of integration".
 
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HallsofIvy said:
Don't forget the "constant of integration".
Thaanks
 
Try re-writing the original expression into something more manageable.
 

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