Understanding the Relationship between Differential Equations and Integration

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SUMMARY

The discussion centers on the relationship between differential equations and integration, specifically addressing the fundamental theorem of calculus. Participants clarify that if \( s = \int f(x) \, dx \), then it is indeed true that \( \frac{ds}{dx} = f(x) \). This relationship is foundational in calculus, linking integration and differentiation directly. The confusion expressed by users highlights the importance of understanding these core concepts in mathematical analysis.

PREREQUISITES
  • Understanding of basic calculus concepts, including differentiation and integration.
  • Familiarity with the fundamental theorem of calculus.
  • Knowledge of notation used in calculus, such as \( ds/dt \) and \( \int f(x) \, dx \).
  • Ability to manipulate and interpret mathematical expressions and equations.
NEXT STEPS
  • Study the fundamental theorem of calculus in detail.
  • Explore examples of differential equations and their solutions.
  • Learn about the applications of integration in real-world problems.
  • Practice solving problems that involve both differentiation and integration.
USEFUL FOR

Students of mathematics, educators teaching calculus, and anyone seeking to deepen their understanding of the relationship between differential equations and integration.

haoku
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If ds = integration of f(t) dt then can I say:
ds/dt= f(t)?

I am very confuse with that. please help
 
Last edited:
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I'm very confused too with what you're asking.

It *sounds* like you meant:

If [tex]s = \int f(x) dx[/tex], is it true that [tex]\frac{ds}{dx} = f(x)[/tex]?

If that's not what you mean, please correct me. If it is what you meant, then it's true. It is essentially the fundamental theorem of calculus.
 
Sorry I have corrected and thanks for answering!
 
Last edited:

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