Integration Techniques and Examples

This page delves into advanced integration techniques, focusing on integration by substitution, and provides several worked examples to illustrate these methods.

Definition: Integration by substitution is a method used to evaluate definite integrals by changing the variable of integration.

The general formula for integration by substitution is:

∫[a to b] f(g(x))g'(x) dx = ∫[g(a) to g(b)] f(u) du

Where u = g(x) is the substitution.

The page presents three detailed examples of definite integral calculations:

1. ∫[0 to π] x cos(x) dx
2. ∫[0 to 2] x√(4-x²) dx
3. ∫[0 to π/2] cos³(x) dx

Example: For ∫[0 to π] x cos(x) dx, the solution uses integration by parts: Result: [x sin(x)]₀ᵖⁱ - ∫[0 to π] sin(x) dx = π sin(π) - (-cos(π) + cos(0)) = 0 + 2 = 2

Highlight: The second example, ∫[0 to 2] x√(4-x²) dx, demonstrates the power of substitution in simplifying complex integrals.

The page concludes with a note on the practical applications of definite integrals, emphasizing their use in calculating areas and volumes in various fields of science and engineering.

Vocabulary: Całka Riemanna wzór (Riemann integral formula) is the mathematical expression that defines the definite integral as the limit of Riemann sums.

This comprehensive guide provides students with a solid foundation in understanding and applying definite integrals, answering questions like "Co mierzy całka?" (What does an integral measure?) and "Do czego służy rachunek całkowy?" (What is the purpose of integral calculus?).