Introduction to Integral Calculus

This page introduces the concept of integral calculus and its relationship to differentiation. It explains the fundamental idea of finding antiderivatives, also known as indefinite integrals.

Definition: Integration is the process of calculating integrals, which is the reverse operation of differentiation.

The page illustrates the connection between derivatives and antiderivatives, showing how integrating (or "anti-differentiating") a function leads to its antiderivative.

Example: For the function f(x) = 2x, the antiderivative F(x) = x² + C, where C is a constant.

The concept of unbestimmte Integrale (indefinite integrals) is introduced, emphasizing that there are infinitely many antiderivatives for a given function, differing only by a constant.

Highlight: The notation for indefinite integrals is ∫f(x)dx = F(x) + C, where C is an arbitrary constant of integration.

Rules for Definite Integrals

This final page covers important rules and properties specific to bestimmte Integrale (definite integrals), enhancing the understanding of their behavior and calculation methods.

Key rules presented include:

1. Zero integral when upper and lower limits are the same
2. Interval additivity
3. Sign change when swapping integration limits
4. Constant multiple rule
5. Sum rule

Example: The interval additivity rule is demonstrated: ∫[a to b] f(x)dx + ∫[b to c] f(x)dx = ∫[a to c] f(x)dx

These rules are crucial for simplifying complex definite integral calculations and understanding the properties of definite integrals.

Highlight: The sign change rule, ∫[a to b] f(x)dx = -∫[b to a] f(x)dx, is particularly useful when dealing with integrals where the limits need to be swapped.

The page concludes with practical examples applying these rules, reinforcing their application in solving definite integral problems.

Vocabulary: Intervalladditivität (Interval Additivity) - A key property of definite integrals that allows breaking down integrals over larger intervals into sums of integrals over smaller intervals.