Understanding Integration Constants: Debunking Common Misconceptions

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SUMMARY

This discussion clarifies the role of integration constants in calculus, specifically addressing misconceptions about definite and indefinite integrals. When integrating a function with limits, such as in the example provided, the constant of integration is not included. However, in indefinite integration, a constant (denoted as A) is essential, as it represents the family of functions that differ by a constant. The example given illustrates that knowing only the derivative of a function limits the understanding of the function itself to an additive constant.

PREREQUISITES
  • Understanding of single-variable calculus
  • Familiarity with definite and indefinite integrals
  • Knowledge of integration constants and their significance
  • Basic proficiency in mathematical notation and functions
NEXT STEPS
  • Study the Fundamental Theorem of Calculus for deeper insights into integration
  • Explore examples of definite vs. indefinite integrals in various contexts
  • Learn about the implications of integration constants in differential equations
  • Review advanced integration techniques, such as integration by parts and substitution
USEFUL FOR

Students and educators in mathematics, particularly those studying calculus, as well as anyone looking to clarify their understanding of integration concepts and constants.

wumple
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Hi,

I thought that if you integrate with limits, you don't include a constant, but if you don't integrate with limits (indefinite), there is a constant. But my book gives the example (all functions are single variable functions, initially of x but then changed to s for the integration):

f' = \frac{1}{2}(\phi'+\frac{\psi}{c})

Integrating:

f(s) = \frac{1}{2}\phi(s) + \frac{1}{2c}\int_0^s\psi + A

What's going on here?
 
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the first statement implies the second. I.e. if all you know about f is its derivative, then you can only know f up to an additive constant.

try to get away from memorizing mindless rules like the (flawed) ones you stated. learn what the concepts mean.
 

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