# **Exercise Questions** All questions in this exercise are listed below. Click on a question to view its solution. --- --- # **Key Concepts** This exercise focuses on the following concepts: * **Antiderivatives:** Finding a function whose derivative is the given integrand. * **Indefinite Integrals:** Representing the family of all antiderivatives using the integral symbol and the constant of integration. * **Power Rule for Integration:** Applying the rule to variables raised to any real power (except -1). * **Exponential Integration:** Integrating functions where the variable is in the exponent, specifically with bases other than $e$. * **Linearity of Integrals:** Using the sum, difference, and constant multiple rules to break down complex integrands. --- # **Important Formulas** Below are the key formulas used in this exercise: | Rule Name | Formula | | :--- | :--- | | **Power Rule** | $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$ | | **Exponential Rule (Base $a$)** | $\int a^x \, dx = \frac{a^x}{\ln a} + C$ | | **Constant Multiple Rule** | $\int k f(x) \, dx = k \int f(x) \, dx$ | | **Sum/Difference Rule** | $\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx$ | --- # **Summary** This exercise covers the fundamental techniques of indefinite integration. The primary focus is on mastering the **Power Rule** for various forms of $x^n$ (including radicals and fractions) and the **Exponential Rule** for general bases. A common strategy throughout these problems is to simplify the integrand algebraically—such as converting radicals to fractional exponents or distributing terms—before applying integration rules. For more complex forms, substitution methods are introduced to align the integrand with standard rules. Always remember to include the **constant of integration ($C$)** for every indefinite integral.