Partial Fraction Decomposition problem

In summary, the conversation discusses the evaluation of the integral ∫((secx)^2)/[((tanx)^2)+(3tanx)+2] using partial fraction decomposition. One person shares their attempt at solving it, but is unsure why their method is incorrect. Another person points out an error in the other solution and offers to help with another question in a new thread.
  • #1
sashab
12
0

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([itex]\frac{1}{2tanx+3}[/itex]+[itex]\sqrt{4(tanx+3/2)^{2}-1}[/itex]), 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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  • #2
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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  • #3
Ohh okay, I'm glad it's not incorrect then! Here's the other solution I was referring to :
tumblr_n3y5ulxl4Q1tsd2vco1_1280.png
 
  • #4
There's an error here: csc = 1/sec.
 
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  • #5
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?
 
  • #6
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.
 

1. What is Partial Fraction Decomposition?

Partial Fraction Decomposition is a method used to simplify and break down a rational expression into smaller, simpler fractions. It is commonly used in calculus and algebra to solve integrals and other mathematical problems.

2. When is Partial Fraction Decomposition used?

Partial Fraction Decomposition is used when we have a rational expression with a polynomial in the numerator and denominator. It is also used in the process of solving integrals, as it helps to break down the integrand into easier terms.

3. What are the steps involved in Partial Fraction Decomposition?

The steps involved in Partial Fraction Decomposition are as follows:

  • 1. Factor the denominator of the rational expression.
  • 2. Write the rational expression as a sum of simpler fractions, with each fraction having a distinct denominator.
  • 3. Set up a system of equations using the coefficients of the terms in the original rational expression.
  • 4. Solve the system of equations to find the unknown coefficients.
  • 5. Combine the simpler fractions into one expression.

4. What are some common mistakes made in Partial Fraction Decomposition?

Some common mistakes made in Partial Fraction Decomposition include:

  • 1. Not properly factoring the denominator of the rational expression.
  • 2. Mixing up the signs of the terms in the system of equations.
  • 3. Forgetting to include all the terms in the original rational expression when setting up the system of equations.
  • 4. Making errors in solving the system of equations.

5. How can Partial Fraction Decomposition be useful?

Partial Fraction Decomposition can be useful in simplifying and solving complex rational expressions, as well as in solving integrals. It can also help in understanding the behavior of a function and identifying vertical and horizontal asymptotes. In addition, it is a useful tool in engineering and physics for solving problems involving transfer functions.

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