1. ## integration constant

why do we put arbitrary constant in integration

2. Differentiate these functions

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

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

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

What do you get? What does this mean?

3. ## Hey there...

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

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

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

6. Originally Posted by pickslides
. Thank you.