Integration by Parts Problem (Natural Log)

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SUMMARY

The forum discussion focuses on solving the integral of ln(x² + 4) using integration by parts. The user sets u = ln(x² + 4) and dv = dx, leading to the expression x ln(x² + 4) - ∫(2x²/(x² + 4))dx. A key suggestion is to simplify the integral by rewriting it as 2 ∫(1 - 1/(1 + (x/2)²))dx, which facilitates the integration process. The discussion emphasizes the importance of recognizing the integral of 1/(1 + x²) for completing the solution.

PREREQUISITES
  • Understanding of integration techniques, specifically integration by parts.
  • Familiarity with logarithmic functions and their properties.
  • Knowledge of trigonometric integrals, particularly ∫(1/(1 + x²))dx.
  • Basic algebraic manipulation skills for simplifying integrals.
NEXT STEPS
  • Study the method of integration by parts in detail, focusing on its applications.
  • Practice integrating logarithmic functions, particularly ln(x² + a).
  • Learn how to manipulate integrals involving rational functions for simplification.
  • Explore trigonometric integrals and their derivations, especially ∫(1/(1 + x²))dx.
USEFUL FOR

Students studying calculus, particularly those tackling integration problems, as well as educators looking for effective teaching methods for integration techniques.

abel216
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Homework Statement


[Intgrl]ln(x^(2)+4)dx

Homework Equations


[Intgrl]udv=uv-[Intgrl]vdu

The Attempt at a Solution


[Intgrl]ln(x^(2)+4)dx, u=ln(x^(2)+4), du=(2x/x^(2)+4), dv=dx, v=x
xln(x^(2)+4)-[Intgrl](2x^(2)/(x^(2)+4))dx
 
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You're almost done. Just write
[tex]\int \frac{2x^2}{x^2 + 4} dx = 2 \int \left( 1 - \frac{1}{1 + \left(\frac{x}{2}\right)^2} \right) dx[/tex]
and substitute [tex]t = \frac{x}{2}[/tex]. (You do know how to integrate [tex]\frac{1}{1 + x^2}[/tex], right?)
 

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