Dealing with functions of several variables

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SUMMARY

The integral problem presented involves evaluating the expression \(\int_0^1 \frac{t^x - t^y}{\ln t} dt\) under the conditions \(x > -1\) and \(y > -1\). The discussion highlights the challenges faced by students when encountering integrals related to functions of several variables, particularly when the relevant chapter has been omitted from the curriculum. Integration by parts (IBP) is suggested as a potential method to tackle the problem, emphasizing the need for familiarity with advanced calculus techniques.

PREREQUISITES
  • Understanding of integral calculus, specifically improper integrals.
  • Familiarity with integration by parts (IBP) technique.
  • Knowledge of logarithmic functions and their properties.
  • Concepts related to functions of several variables, such as \(f(x,y,z)\).
NEXT STEPS
  • Study the method of integration by parts in detail.
  • Explore the properties of logarithmic integrals.
  • Review advanced calculus topics related to functions of several variables.
  • Practice solving integrals involving parameters and limits.
USEFUL FOR

Students studying calculus, particularly those preparing for exams involving integrals and functions of several variables, as well as educators looking for examples of complex integral problems.

Nick89
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Homework Statement


Assuming [itex]x > -1[/itex] and [itex]y > -1[/itex], find the following integral:
[tex]\displaystyle \int_0^1 \frac{t^x - t^y}{\ln t} dt[/tex]


Homework Equations





The Attempt at a Solution


I have no idea where to start on this one... It came up in an exam for a class dealing with functions of several variables (eg, f(x,y,z)) so I don't think it's a usual substitution problem...

I tried writing it as [tex]\int t^x \ln(-t) dt[/tex] (minus the same for y) but I couldn't get any further...

Do I perhaps have to find some taylor polynomial or something?
 
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Nick89 said:
I tried writing it as [tex]\int t^x \ln(-t) dt[/tex] (minus the same for y) but I couldn't get any further...
This is wrong that's not the same integral ( do you know why?)

Use IBP.
 


Sorry I screwed up there, heh.

Anyway, I just found out that this question in the exam was part of some chapter that is now omitted, that would explain why I have no idea how to start on this one ^^
 

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