SUMMARY
The integral \(\int^{1}_{0} \sin(\pi x) \sin(n \pi x) dx\) evaluates to 1 when \(n = 1\) and 0 for all integers \(n \geq 2\). This result is derived from the orthogonality property of sine functions over the interval [0, 1]. Specifically, the trigonometric addition formulas facilitate the simplification of the integral, confirming the stated outcomes based on the value of \(n\).
PREREQUISITES
- Understanding of integral calculus
- Familiarity with trigonometric functions
- Knowledge of orthogonality in function spaces
- Basic proficiency in using trigonometric addition formulas
NEXT STEPS
- Study the properties of orthogonal functions in Fourier series
- Explore the derivation of trigonometric addition formulas
- Learn about the applications of integrals in signal processing
- Investigate the implications of sine function orthogonality in physics
USEFUL FOR
Mathematicians, physics students, educators, and anyone interested in the applications of trigonometric integrals and orthogonality in mathematical analysis.