Originally Posted by

**derfleurer** If you change C, you change the function. $\displaystyle x + 4$ and $\displaystyle x + 3$ may have the same derivative, but for all x, the two functions are completely different. Your constant of integration is just that; a constant. Constants in no way affect your derivative and so your +C can be anything from $\displaystyle ln5$ to $\displaystyle \pi$.

We don't always take the integral from b to a. That's only when we're finding accumulated area, in which case the +C is negligable. But sometimes we just want F(a) or F(b), in which case it's very important to know.

Say we know the antiderivative of $\displaystyle f(x)$ is $\displaystyle x^3 + 4x + C$. And say we know that $\displaystyle F(0) = 1$. This gives us a point on the function with which to calculate C. We know that $\displaystyle (0)^3 + 4(0) + C = 1$, so C must equal 1.