Integrating a Definite Integral with a Dummy Variable

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SUMMARY

The discussion centers on the integration of a definite integral with a dummy variable, specifically the integral f(x) = ∫sint dt from 0 to x. Participants clarify that substituting the dummy variable t with the upper limit x during integration is incorrect. The correct evaluation of the integral yields -cos(x) after applying the Fundamental Theorem of Calculus. The consensus emphasizes that while it may appear to work in some cases, this substitution method is not generally valid.

PREREQUISITES
  • Understanding of definite integrals and dummy variables in calculus
  • Familiarity with the Fundamental Theorem of Calculus
  • Basic knowledge of trigonometric functions and their integrals
  • Experience with evaluating integrals with specific limits
NEXT STEPS
  • Study the Fundamental Theorem of Calculus in detail
  • Practice evaluating definite integrals with various functions
  • Learn about common pitfalls in integration techniques
  • Explore the properties of dummy variables in calculus
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Students and educators in mathematics, particularly those focusing on calculus, as well as anyone looking to deepen their understanding of integration techniques and the use of dummy variables.

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If we have a definite integral given by f(x) = ∫sintdt with limits from 0 to x, can we integrate the function directly by substituting t as x because the variable involved in the integrand is a dummy variable?
 
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What do you mean by "substituting t as x"? Certainly after the integration, you have
[tex]\int_0^x sin(t)dt= cos(t)\right]_0^x= -cos(x)[/tex]
 
Do you mean set t = x and take the anti-derivative and forget about the limits of integration? No. It might work by accident in some problems, but in general, you can't do that.

Consider [tex]\int_0^x e^t dt[/tex]
 

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