I with integration using partial fractions

A, B, and C are constants to be determined. In summary, the integral can be simplified using partial fractions, but the initial attempt may have been incomplete by not including the quadratic term.
  • #1
farisallil
1
0

Homework Statement



Compute the integral:
int ((1-x^2)/(x^3+x)) dx



Homework Equations


int ((1-x^2)/(x^3+x)) dx


The Attempt at a Solution


I think I should use the partial fraction method to simplify the fraction
so
(1-x^2)/(x^3+x)= A/x + B/ (1+x^2)
Therefore
A(1+x^2)+B(x)=1-x^2
putting it in a polynomial form:
x^2 (A)+x (B) + 1 (A)= 1- x^2
and by equating the coefficients,
A = -1
B = 0
However, A doesn't satisfy the constant on the LHS (+1)

So what is the thing that I did wrong??

Thanks in advance
 
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  • #2
I think you missed the quadratic part of your partial fraction decomposition, it should be

[tex] \frac{1-x^2}{x(1+x^2)}= \frac{A}{x}+\frac{B+Cx}{1+x^2} [/tex]
 

FAQ: I with integration using partial fractions

What is "I with integration using partial fractions"?

"I with integration using partial fractions" refers to the process of breaking down a complex fraction into simpler fractions using the method of partial fractions. This method is used in integration to make it easier to solve complex integrals.

Why is partial fractions used in integration?

Partial fractions are used in integration because it helps to simplify complex integrals into smaller, more manageable parts. This makes it easier to solve the integral and obtain the final answer.

How does the method of partial fractions work?

The method of partial fractions involves breaking down a complex fraction into simpler fractions with known denominators. This is done by equating the original fraction to an equation with unknown variables, and then solving for those variables through a system of linear equations.

What are the steps involved in using partial fractions in integration?

The steps involved in using partial fractions in integration are as follows:

  • 1. Factorize the denominator of the complex fraction into linear and quadratic factors.
  • 2. Write the original fraction as a sum of simpler fractions with the corresponding factors as denominators.
  • 3. Equate the original fraction to the sum of the simpler fractions.
  • 4. Solve for the unknown variables by setting up and solving a system of linear equations.
  • 5. Integrate each simpler fraction separately and combine the results to obtain the final answer.

What are the most common mistakes when using partial fractions in integration?

Some common mistakes when using partial fractions in integration include:

  • Forgetting to include all the possible factors in the partial fractions decomposition.
  • Incorrectly equating the original fraction to the sum of simpler fractions.
  • Making errors in solving the system of linear equations to find the unknown variables.
  • Forgetting to include the constant of integration when integrating each simpler fraction.

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