Definite integral with x^2+c in the denominator

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SUMMARY

The discussion focuses on solving the definite integral \(\int_{2.6}^{5.5} \frac{1}{x^2+9}dx\). Participants clarify that the initial attempt using logarithmic functions is incorrect, as the derivative of \(\ln(x^2+9)\) does not yield the integrand. A trigonometric substitution is necessary to correctly evaluate this integral, specifically using the substitution \(x = 3\tan(\theta)\) to simplify the expression.

PREREQUISITES
  • Understanding of definite integrals and their properties
  • Familiarity with trigonometric substitutions in calculus
  • Knowledge of derivatives, particularly the chain rule
  • Basic skills in manipulating logarithmic functions
NEXT STEPS
  • Study trigonometric substitution techniques for integrals
  • Learn how to derive integrals involving rational functions
  • Practice evaluating definite integrals with variable limits
  • Explore the application of the chain rule in calculus
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Students studying calculus, particularly those working on integral calculus, and educators looking for examples of trigonometric substitution in definite integrals.

Hypatio
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Homework Statement

Homework Equations



solve the definite integral

[tex]\int_{2.6}^{5.5} \frac{1}{x^2+9}dx[/tex]

The Attempt at a Solution



ln(5.5^2+9)-ln(2.6^2+9) doesn't seem correct
 
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Can you show your steps in getting that solution? What did you get for the antiderivative of the integrand?
 
Hypatio said:

Homework Statement




Homework Equations



solve the definite integral

[tex]\int_{2.6}^{5.5} \frac{1}{x^2+9}dx[/tex]

The Attempt at a Solution



ln(5.5^2+9)-ln(2.6^2+9) doesn't seem correct

If you take the derivative of log(x^2+9) you will not get 1/(x^2+9) (at least if you don't forget the chain rule). You need a trig substitution to do that integral.
 

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