A rookie question for integrals of polynomial functions

[YoungPhysicist]

$$\int x^2+3 = \frac{x^3}{3}+3x+C$$

I can get the front two part by power rule, but what is the C doing there? Wolframalpha suggested it should be a constant, but what value should it be? Sorry for asking rookie questions

 

[PeroK]

$$\int x^2+3 = \frac{x^3}{3}+3x+C$$

I can get the front two part by power rule, but what is the C doing there? Wolframalpha suggested it should be a constant, but what value should it be? Sorry for asking rookie questions

$C$ could be any constant. Whatever it is, it vanishes when you differentiate.

This means that technically the antiderivative (indefinite integral) is not a single function, but a set (actually an equivalence class) of functions. If you have a boundary condition, the required value at $x=0$, say, then you can calculate $C$ and you have a specific function.

Question for you: why does $C$ not matter for a definite integral? E.g.
$$\int_1^4 x^2+3 dx$$

 

[RPinPA]

This means that technically the antiderivative (indefinite integral) is not a single function, but a set (actually an equivalence class) of functions.

Specifically, the set of functions which are displaced up and down the y-axis relative to each other. If you take any function and shift it up or down, you don't change its slope. So the arbitrary constant C is an arbitrary, usually real, number indicating any vertically-shifted version of this same function.

I say usually because sometimes you're dealing with complex-valued functions and then C is any complex number.

 

[YoungPhysicist]

Question for you: why does CCC not matter for a definite integral? E.g.

∫41x2+3dx​

$$\int _1^4 x^2 +3 dx =$$
$$\frac{4^3}{3}+3(4)+C-\frac{1^3}{3}-3(1)-C$$
$$=30$$
So the $C$ just cancels out.