# A rookie question for integrals of polynomial functions

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

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

Homework Helper
Gold Member
2022 Award
$$\int x^2+3 = \frac{x^3}{3}+3x+C$$

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.
$$\int_1^4 x^2+3 dx$$

Delta2, YoungPhysicist and QuantumQuest
Homework Helper
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.

Delta2
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.

PeroK