Confusion about the constant of integration and bounds

In summary, the constant of integration is a term used when integrating a function that represents the family of curves that could fit the given function. It is important because it allows for a more general integration process. The constant of integration cannot be determined without additional information and can be positive, negative, or zero. It is denoted by "C" and can be calculated by using initial values or boundary conditions. The constant of integration is added to the integrated function to account for all possible solutions within the given bounds, also known as the limits of integration. It does not change the shape of the integrated function, but only adds a vertical shift on the y-axis by a fixed amount.
  • #1
RandomMystery
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When we integrate a derivative:
gif.latex?\frac{dy}{dx}=&space;x^{5}+x^{4}+x^{3}.gif

What are the bounds?
gif.latex?\int_{?}^{?}dy=&space;\int_{?}^{?}x^{5}+x^{4}+x^{3}dx.gif

Shouldn't we then have this if the bounds are x2 and x1:
}{6}+\frac{x_{1}^{5}}{5}+\frac{x_{1}^{4}}{4}&space;\right&space;]&space;+&space;C.gif

but instead we have this:
f.latex?y=&space;\frac{x_{_{2}}^{6}}{6}+\frac{x_{2}^{5}}{5}+\frac{x_{2}^{4}}{4}+C.gif
 

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  • gif.latex?\frac{dy}{dx}=&space;x^{5}+x^{4}+x^{3}.gif
    gif.latex?\frac{dy}{dx}=&space;x^{5}+x^{4}+x^{3}.gif
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  • #2
The constant of integration is a way of acknowledging that there are infinitely many solutions for every indefinite integral. When we integrate any function from [tex]x_1[/tex] to [tex]x_2[/tex] we are finding the area beneath the curve. Adding on a constant C would no longer give us the correct area beneath the curve, provided that C is not equal to zero.
 

1. What is the constant of integration and why is it important?

The constant of integration is a term that is used when integrating a function. It is added at the end of the integration process and represents the family of curves that could fit the given function. It is important because it allows for the integration process to be more general and account for all possible solutions.

2. How do I determine the constant of integration?

The constant of integration cannot be determined without additional information or conditions. It is usually denoted by "C" and can be calculated by using initial values or boundary conditions.

3. Can the constant of integration be negative?

Yes, the constant of integration can be positive, negative, or zero. It depends on the function being integrated and the specific conditions or information given.

4. What are bounds and how do they relate to the constant of integration?

Bounds refer to the limits of integration, which can be either a specific value or a range. The constant of integration is added to the integrated function to account for all possible solutions within the given bounds.

5. Can the constant of integration change the shape of the integrated function?

No, the constant of integration does not change the shape of the integrated function. It only adds a vertical shift to the function, shifting it up or down on the y-axis by a fixed amount.

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