# Math Help - integration constant

1. ## integration constant

why do we put arbitrary constant in integration

2. Differentiate these functions

$y_1 = x^2+10x+7$

$y_2 = x^2+10x-9$

$y_3 = x^2+10x+2$

What do you get? What does this mean?

3. ## Hey there...

It's...
The fundamental theorem of calculus:

$F(x)=\int_a^x{f(t)dt}$
where $f$ is a real-valued function continuous on $[a, b]$. Then, $F$ is continuous on $[a, b]$, differentiable on the open interval $(a, b)$, and:
$F'(x)=f(x)$ $\\\\$ $\forall x \in (a,b)$

4. Originally Posted by prasum
why do we put arbitrary constant in integration
Say you want to integrate $f(x) = x^2$. Writing $\int{2x}\;{dx} = x^2$ means to say that the function $f(x) = 2x$ satisfies $F'(x) = x^2$. But is it the only one that satisfies this relation? Clearly, $2x+1$ satisfies; so do $2x+2$ and $2x+3$. In fact, for any number $C$, it's true that $2x+C$ satisfies the relation. Finding $\int{f(x)}\;{dx}$ means finding the set of all function which satisfy F'(x) = f(x), i.e the set of all anti-derivatives of f(x).

5. Originally Posted by TheCoffeeMachine
Writing $\int{x^2}\;{dx} = 2x$

6. Originally Posted by pickslides
. Thank you.