Advanced Integration Techniques
This page delves into more complex integration techniques, focusing on functions involving roots, higher powers, and compound expressions. It provides examples of Stammfunktion bilden Übungen (antiderivative formation exercises) with step-by-step solutions.
The document starts with examples of integrating functions containing square roots:
Example: For f(x) = √x, the antiderivative is F(x) = 2/3x^(3/2).
It then progresses to more complex functions, including those with higher powers and negative exponents:
Example: For f(x) = x^(-1/2), the antiderivative is F(x) = 2√x.
The page also covers the integration of functions involving parentheses and multiple terms:
Example: For f(x) = 2x(-x² + 6x + 5), the antiderivative is F(x) = -1/2x⁴ + 4x³ + 5x² + c.
The document includes examples of Bestimmtes Integral berechnen (calculating definite integrals), demonstrating the fundamental theorem of calculus:
Highlight: The definite integral is calculated by evaluating the antiderivative at the upper and lower limits and subtracting.
Several practice problems are provided, allowing students to apply the Stammfunktion Regeln (antiderivative rules) they've learned. These exercises cover a range of function types, from simple polynomials to more complex expressions involving roots and trigonometric functions.
Vocabulary: Limes (limit) is used in some of the more advanced integration problems.
The page concludes with some notes on integration techniques for specific types of functions, reinforcing the importance of recognizing patterns and applying the appropriate Stammfunktion Regeln.
