# Why does the indefinite integral have a constant?

## Question:

Why does the indefinite integral have a constant?

## Indefinite Integrals:

An indefinite integral is one where we integrate a function but there are no limits of integration. In an indefinite integral, our final answer would then be in the form of a function since we are not actually integrating any values.

We will let the functions:

• {eq}\displaystyle \rm f(x) = x^2 + 1 {/eq}
• {eq}\displaystyle \rm g(x) = x^2 {/eq}

Now if we take the derivative of each function, we will have:

• {eq}\displaystyle \rm \frac{df}{dx} = 2x {/eq}
• {eq}\displaystyle \rm \frac{dg}{dx} = 2x {/eq}

Now even if the functions are not the same, their derivatives are because the difference is only a constant. So here if we integrate our derivative,

{eq}\displaystyle \rm f(x) = \int\ 2x\ dx {/eq}

We will never be able to get our function f(x) here unless we add our constant of integration. This is the reason why we add our constant of integration, because each indefinite integral actually has an infinite amount of solutions due to each and every constant having a zero derivative. Here we can integrate our function as:

{eq}\displaystyle \rm f(x) = x^2 + C {/eq}

C here can thus take any value of a constant, we can substitute C = 1 here to get our original f(x) or just C = 0 to get our original g(x). 