Partial Fraction Decomposition problem

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Homework Help Overview

The problem involves evaluating the integral of a rational function that includes secant and tangent functions, specifically ∫((secx)^2)/[((tanx)^2)+(3tanx)+2]. The context centers around partial fraction decomposition as a method of integration.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss the original poster's attempt at solving the integral and express confusion regarding the correctness of their method compared to an alternative solution. Questions arise about the behavior of the functions involved and the validity of the different approaches.

Discussion Status

The discussion is active, with participants sharing their thoughts on the methods used and questioning the assumptions behind the solutions. Some participants express agreement with the original poster's approach, while others point out potential errors in reasoning. There is a willingness to explore further questions related to partial fraction decomposition.

Contextual Notes

Participants note discrepancies in the behavior of the functions derived from the integral and discuss the implications of these differences. There is mention of a need for clarity regarding the correctness of various methods used in the problem.

sashab
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Homework Statement



Evaluate ∫((secx)^2)/[((tanx)^2)+(3tanx)+2]

Homework Equations



Partial fraction decomposition

The Attempt at a Solution



So here's what I did:
tumblr_n3y124QYZc1tsd2vco1_500.jpg


But this is incorrect. It says the correct answer is -2lnabs(\frac{1}{2tanx+3}+\sqrt{4(tanx+3/2)^{2}-1}), which was achieved by making the same u-substitution, completing the square, and then doing a trig substitution. I'm not sure why my method is incorrect. Any help would be great, thanks!
 
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I agree with your answer (and method!). Seems to me that the other answer is a decreasing function in the range 0 to 1, yet from the integral it should clearly be increasing. If you post all the working of the other method I'll see if I can find the error.
 
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Ohh okay, I'm glad it's not incorrect then! Here's the other solution I was referring to :
tumblr_n3y5ulxl4Q1tsd2vco1_1280.png
 
There's an error here: csc = 1/sec.
 
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haruspex said:
There's an error here: csc = 1/sec.
Ohh okay I see. Thanks so much for the help! Would you be willing to help me with another partial fraction decomposition question?
 
sashab said:
Ohh okay I see. Thanks so much for the help! Would you be willing to help me with another partial fraction decomposition question?
Sure, but post it in a new thread for all to see.
 

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