The antiderivative (Ger. Stammfunktion)
Every differentiable function $F$, where
$F'(x) = f(x)$,
is called the antiderivative of $f$. And, $f$ is the derivative of $F$.
There is an infinite set of antiderivatives of a function $f(x)$ which we show by adding the constant (Integrationskonstante) $C$ to $F(x)$:
Indefinite integral:
$$\int f(x)\ dx = F(x) + C$$
initial value problem - das Anfangswertproblem
Sometimes you are looking for specific antiderivative restricted by boundary conditions.
This is generally solved by finding a suitable integration constant
example
The function $f(x)= x$. Find the antiderivative F of f that passes through the point $P(1 , \frac{3}{2})$