Integrating by Parts: Showing $\int \frac{1}{1-x^2}dx$

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SUMMARY

The integral of the function $\int \frac{1}{1-x^2}dx$ can be expressed using integration by parts as $\frac{x}{1-x^2}-\int \frac{2x^2}{(1-x^2)^2}dx$. The discussion emphasizes the importance of identifying the correct functions for \( u \) and \( v \) in the integration by parts formula. Participants suggest experimenting with both possibilities for \( u \) and \( v \) to determine the most effective approach for solving the integral.

PREREQUISITES
  • Understanding of integration by parts
  • Familiarity with the integral of rational functions
  • Knowledge of basic calculus concepts
  • Ability to manipulate algebraic expressions
NEXT STEPS
  • Practice integration by parts with different functions
  • Explore the properties of rational functions in calculus
  • Learn about the convergence of improper integrals
  • Investigate advanced techniques for solving integrals, such as substitution methods
USEFUL FOR

Students studying calculus, particularly those focusing on integration techniques, as well as educators looking for examples of integration by parts applications.

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Homework Statement



By integrating by parts , show that

[tex]\int \frac{1}{1-x^2}dx=\frac{x}{1-x^2}-\int \frac{2x^2}{(1-x^2)^2}dx[/tex]

Homework Equations





The Attempt at a Solution



I don see which is u and v.
 
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You can view the first integral as a product of two functions, which ones? Now if you don't see which one should be u and which one should be v just try one. After all there are only two possibilities.
 
Perhaps it would further help to think about how the x could appear in the numerator of the first term on the right hand side.
 

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