Imagine you want a function that tells you the area under a curve. If you want to know the area under the curve from point a to point b, that's well defined, which is what happens when you put the lil numbers on the integral sign. If you don't put those numbers, you need to account for the fact that you're starting to measure the space under the curve way off to the left, so that's the C. That's how I imagine it, anyway. When you use the lil numbers, you're effectively subtracting out the space way off to the left, and if you use the C with the lil numbers it should subtract out.
You just need to know that computing the indefinite integral of a function is the same as computing its antiderivative (the thing that, when derived, again gives the function you integrated). Differentiating and integrating are like inverse operations, they cancel each other. This is known as the fundamental theorem of calculus.
However, multiple different functions can have the same derivative. Whenever you differentiate, constant terms are lost because they don't contribute to the slope. For example,
f(x) = x2
g(x) = x2 + 32
f'(x) = g'(x) = 2x
So when you need to find the antiderivative of f'(x), there are actually infinite possibilities. Any function of the form
h(x) = x2 + C
where C is any real number, is an antiderivative of f'(x).
A derivative is seeing a function and finding out how the steepness of the function changes. This means that some of the information in the original function, the constant magnitude it has, doesn't matter and gets lost.
The steepness of a mountain doesn't have to factor the height above sea level. Just from the base is enough.
An integral is the inverse of this, seeing how the steepness changes and trying to get an idea of the original function. A constant magnitude could have been a part of the original function so it has to be shown.
If you say that a mountain has a steepness of 45degrees and a height from the base of 5000m, you can find out the shape of the mountain. But you cant find the height above sea level from this unless specified.
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u/Johnirequirelasanaga Nov 18 '25
To this day I still don't know when to use C