SUMMARY
The integral of the function \( a^x \) is proven to be \( \frac{a^x}{\ln(a)} + C \). The proof begins by rewriting the integral as \( \int e^{\ln(a) \cdot x} \, dx \). A substitution is then applied where \( u = \ln(a) \cdot x \), facilitating the integration process. This method effectively demonstrates the relationship between exponential functions and logarithms in integration.
PREREQUISITES
- Understanding of integral calculus
- Familiarity with exponential functions
- Knowledge of logarithmic properties
- Experience with substitution methods in integration
NEXT STEPS
- Study the properties of exponential functions and their integrals
- Learn about substitution techniques in integral calculus
- Explore the relationship between logarithms and exponential functions
- Practice solving integrals involving different bases and their transformations
USEFUL FOR
Students of calculus, mathematics educators, and anyone seeking to deepen their understanding of integration techniques involving exponential functions.