Question

Why does do definite integrals not have a C afterwards?

Answer 1

Because the C cancels out when you take the difference between F(a) and F(b)

Answer 2

definite integrals are definite. like a set value. if you add a C it wouldnt be definite ROFL

Answer 3

Due to the Fundamental Thm of Calculus it can be shown that
\[ \int_a^b f(x) dx = F(b) -F(a) \]

Lets take a definite integral, but make its upper bound a new variable, so:
\[ g(t) = \int_a^t f(x) dx = F(t) -F(a) \]

This essentially is an indefinite integral, except a can start anway, so F(a) becomes an unknown constant and F(t) is the anti-derivative in terms of t

Answer 4

you are taking the area of a predefined function, as opposed to trying to find some family of functions as an antiderivative.

Answer 5

* a can start anywhere

Answer 6

You see definite integral have limits and in the end we subtract upper limit-lower limit. This is in the form F(U)+C-(F(L)+C)=F(U)-F(L) .Hence C disappears.

Answer 7

or if we address this as a reimann sum
\[\lim_{n \to inf}~\sum_{i=1}^{n} f(a+i\frac{b-a}{n})\frac{b-a}{n}=\int_{a}^{b}f(x)~dx\]