SUMMARY
The antiderivative of the function (e^sin(t)) * (cos(t)) is indeed e^(sin(t)) + C. The confusion arose from the application of the chain rule, which simplifies the integration process. The cosine term does not appear in the final result due to its role in the derivative of the sine function, effectively canceling out during integration. This illustrates the importance of understanding the chain rule in calculus.
PREREQUISITES
- Understanding of basic calculus concepts, particularly integration.
- Familiarity with the chain rule in differentiation.
- Knowledge of exponential functions and their properties.
- Experience with antiderivatives and indefinite integrals.
NEXT STEPS
- Study the chain rule in depth to understand its application in integration.
- Practice finding antiderivatives of exponential functions with trigonometric components.
- Explore integration techniques involving substitution methods.
- Review examples of integrating products of functions, particularly involving e^x.
USEFUL FOR
Students studying calculus, particularly those focusing on integration techniques, and educators looking for examples of applying the chain rule in antiderivatives.