SUMMARY
The discussion focuses on the differentiation and integration of functions involving inverse sine and polynomial expressions. The derivative of the function \( y = \sin^{-1} \left( \frac{X+a}{b} \right) \) is established as \( \frac{1}{\sqrt{b^2 - x^2 - 2ax - a^2}} \). The integration of \( \frac{x}{\sqrt{9-4x-x^2}} \) leads to a substitution method where \( u = 4x + x^2 \) simplifies the integral. The constants \( a \) and \( b \) are determined to be \( a = 2 \) and \( b = \sqrt{13} \) based on the equations derived from the integration process.
PREREQUISITES
- Understanding of calculus concepts, specifically differentiation and integration.
- Familiarity with inverse trigonometric functions, particularly \( \sin^{-1} \).
- Knowledge of substitution methods in integration.
- Ability to manipulate algebraic expressions involving polynomials.
NEXT STEPS
- Study the properties and applications of inverse trigonometric functions.
- Learn advanced integration techniques, including substitution and integration by parts.
- Explore the use of derivatives in real-world applications, particularly in physics and engineering.
- Investigate polynomial long division and its role in simplifying integrals.
USEFUL FOR
Students and professionals in mathematics, particularly those studying calculus, as well as educators looking for examples of differentiation and integration techniques involving inverse functions.
