Understanding Integration Constants: Debunking Common Misconceptions

In summary, the conversation discusses the concept of integrating with limits and without limits. It is mentioned that when integrating with limits, a constant is not included, but when integrating without limits, a constant is present. The example given involves a single variable function represented by f' and its integration, which results in the function f(s). The speaker questions this example and suggests understanding the concepts rather than memorizing rules. It is concluded that knowing only the derivative of a function allows for a constant to be added to the function.
  • #1
wumple
60
0
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):

[tex] f' = \frac{1}{2}(\phi'+\frac{\psi}{c}) [/tex]

Integrating:

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

What's going on here?
 
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  • #2
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.
 

FAQ: Understanding Integration Constants: Debunking Common Misconceptions

1. What is an integration constant?

An integration constant is a constant that is added to the solution of a differential equation after integrating. It accounts for any unknown initial conditions or other factors that may affect the solution.

2. Why is an integration constant necessary?

An integration constant is necessary because when we integrate a differential equation, the result is a family of curves rather than a single curve. The integration constant allows us to choose the specific curve that satisfies the given initial conditions or other constraints.

3. How do you determine the value of the integration constant?

The value of the integration constant is determined by using the given initial conditions or other constraints. Substituting these values into the solution of the differential equation will give a specific value for the integration constant.

4. Can the value of the integration constant change?

Yes, the value of the integration constant can change if the initial conditions or other constraints are altered. This will result in a different solution curve for the differential equation.

5. How does the integration constant affect the solution of a differential equation?

The integration constant can shift the solution curve up or down, depending on its value. It can also affect the shape of the curve, as it determines the specific solution that satisfies the given initial conditions or other constraints.

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