How do I integrate u^-1 with new limits after substitution?

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SUMMARY

The discussion centers on integrating the function u-1 after performing a substitution in calculus. The user initially struggles with the integration limits and the form of the function but ultimately concludes that the integral of u-1 from 1 to 2 results in ln(2). The conversation highlights the importance of recognizing that u-1 can be integrated using the natural logarithm, despite initial misconceptions about the necessity of a linear expression.

PREREQUISITES
  • Understanding of basic calculus concepts, specifically integration.
  • Familiarity with substitution methods in integration.
  • Knowledge of the natural logarithm and its properties.
  • Ability to manipulate limits of integration after substitution.
NEXT STEPS
  • Study the properties of the natural logarithm, particularly its application in integration.
  • Learn about integration techniques involving improper integrals and their limits.
  • Explore advanced substitution methods in calculus for more complex integrals.
  • Practice integrating functions of the form u-n for various values of n.
USEFUL FOR

Students studying calculus, particularly those learning integration techniques, and educators seeking to clarify concepts related to logarithmic integration.

markyp23
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Homework Statement



problem.jpg



Homework Equations



None. Well, dx=du/cosx

The Attempt at a Solution



I've substituted it in, got new values for the limits but I have u^-1 on the bottom and so can't integrate it from my current knowledge. Basically I'm stuck with:

Integration of u^(-1) du with limits of 1 and 2.

Any help appreciated. Apologies for not being able to work the Latex system.
 
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Seriously, you haven't encountered the derivative of the natural logarithm yet?
 
I have but I thought it had to be a linear expression to use the natural log. It did cross my mind though - this is what to do, then?
 
What do you mean by a "linear expression"??
 
Something of the form (ax+b).

Decided to assume that (u) is in that form and so worked it through. Is ln(2) the final answer?
 
Ask yourself:
May we write u=1*u+0?

Sure, the answer is ln2
 
Thanks for your help! :)
 

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