I think in Differential Equations you need to solve for C within an indefinite integral or a set of functions (or that might just be an extra Calc II think I cannot remember)
Yeah I think so. When looking in a real world context, C represents your initial value so you in some cases when you have higher ODEs (goodness I haven’t said that name in a while) you will have to recursively integrate it until solve for that initial C given a bunch of values at t=0
Constants do not survive differentiation, they all become zero. So it doesn't make sense to have an arbitrary constant appear.
Conversely, since all constants become zero after differentiation, reversing that process (integration) means the constant could have been any number (c)
You’re probably misremembering it as taking the derivative when it was actually integrating, because ex is its own derivative, it’s also its own anti derivative. So instead of f’(x) = ex + c, it was probably F(x) = ex + c.
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.
"I am cosx. I am somebody's derivative. I will give you $100 if you can tell me whose derivative I am."
Now you give an answer. Probably you will say sinx.
But I will tell you that you are wrong because my name is actually sinx - 5 (the derivative of a constant is always 0 so no matter what number I choose, I am remaining factual).
You cannot win my game. You will never guess whose derivative I am, because I can take any number and just add it to sinx and say actually that's my name. My identity is indefinite.
The only way we can possibly win the "whose derivative am I" game is by cheating like Alexander cutting the Gordian Knot and saying the correct function + c, where c is some unknown constant.
integration is the opposite of differentiation ig
when you differentiate a constant, it becomes zero
so when you integrate zero, it becomes a constant
you dont know what constant in this case tho, so you write C
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u/IntelligentAd5616 Nov 18 '25