SUMMARY
The integral of the function e^(-3x) / (1 + e^(-3x)) can be solved using u-substitution. By letting u = 1 + e^(-3x), the differential du is calculated as -3e^(-3x) dx, leading to the substitution du/3 = e^(-3x) dx. The final result of the integration is -1/3 ln|1 + e^(-3x)| + C, confirming the correct application of u-substitution in this context.
PREREQUISITES
- Understanding of u-substitution in integration
- Familiarity with exponential functions and their properties
- Basic knowledge of logarithmic functions
- Ability to manipulate differentials in calculus
NEXT STEPS
- Practice additional problems using u-substitution for various integrals
- Explore integration techniques involving exponential functions
- Learn about the properties of logarithmic functions in calculus
- Review the fundamentals of differential calculus to strengthen understanding
USEFUL FOR
Students studying calculus, particularly those focusing on integration techniques, as well as educators looking for examples of u-substitution applications in exponential integrals.