Finding the integral using partial fractions

In summary, to solve the integral x/x^2+4x+13, it is recommended to use the substitution u = x+2 and v = u^2+9. The resulting fractions can then be integrated using the formulas for arctangent and natural logarithm.
  • #1
sdoyle
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0

Homework Statement


Solve the integral x/x^2+4x+13


Homework Equations


I think that you would use partial fractions but I'm not really sure. I know that you need to complete the square on the denominator.


The Attempt at a Solution


The completed square would be (x+2)^2+9. I don't know what to do now b/c of the x term on top. Help!
 
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  • #2
Since the denominator cannot be factored (in terms of real coefficients), there is no "partial fractions".

Let u= x+ 2 so x= u-2. Then the fraction becomes
[tex]\frac{u-2}{u^2+ 9}= \frac{u}{u^2+ 9}-2\frac{1}{u^2+ 9}[/tex]
Let [itex]v= u^2+ 9[/itex] to integrate the first and the second is an arctangent.
 
  • #3
[itex] \frac{x}{x^2+4x+13} = -\frac{2}{x^2+4x+13} + \frac{x+2}{x^2+4x+13} [/itex]

[itex] \int \frac{dx}{x^2+a^2} = \frac{1}{a}ArcTan\frac{x}{a} + C[/itex]

[itex] \int \frac{x'}{x} = Ln|x| + C[/itex]

That's all you need.

Edit: Method below a lot better.
 

What is the concept of finding the integral using partial fractions?

The concept of finding the integral using partial fractions is a method of breaking down a rational function into simpler fractions, making it easier to integrate. It involves finding the partial fraction decomposition of the original function, which is then integrated term by term.

Why is it important to find the integral using partial fractions?

Finding the integral using partial fractions is important because it allows us to solve more complex integrals that cannot be solved using basic integration techniques. It also helps us to express the integral in a simpler form, making it easier to evaluate.

What are the steps involved in finding the integral using partial fractions?

The steps involved in finding the integral using partial fractions are as follows:

  1. Factorize the denominator of the rational function into linear and irreducible quadratic factors.
  2. Write the partial fraction decomposition of the function, where each term has a constant numerator and a factor of its corresponding denominator.
  3. Equate the coefficients of the terms in the partial fraction decomposition to the coefficients of the original function.
  4. Solve for the unknown constants using algebraic manipulation.
  5. Integrate each term in the partial fraction decomposition separately.
  6. Add all the integrals together to obtain the final solution.

What are some common mistakes to avoid when finding the integral using partial fractions?

Some common mistakes to avoid when finding the integral using partial fractions include:

  • Not properly factoring the denominator of the rational function.
  • Forgetting to include all the possible terms in the partial fraction decomposition.
  • Making algebraic errors while solving for the unknown constants.
  • Forgetting to add the constant of integration when integrating each term separately.
  • Not simplifying the final solution before presenting it as the answer.

Can the method of finding the integral using partial fractions be applied to all rational functions?

Yes, the method of finding the integral using partial fractions can be applied to all rational functions. However, it is only useful when the degree of the numerator is less than the degree of the denominator. If the degree of the numerator is equal to or greater than the degree of the denominator, then the rational function cannot be broken down into simpler fractions and the method cannot be applied.

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