# **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: - Integration by substitution (change of variables) - Standard integral forms involving inverse trigonometric functions - Completing the square for quadratic expressions - Integration of composite functions (chain rule in reverse) - Logarithmic integration (derivative over function pattern) --- # **Important Formulas** Below are the key formulas used in this exercise: $$\int \frac{dx}{x^2+a^2} = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C$$ $$\int \frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}\left(\frac{x}{a}\right) + C$$ $$\int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + C$$ $$\int [f(x)]^n f'(x)\,dx = \frac{[f(x)]^{n+1}}{n+1} + C \quad (n \neq -1)$$ Substitution rule: If $u = g(x)$, then $\int f(g(x))g'(x)\,dx = \int f(u)\,du$ --- # **Summary** This exercise demonstrates the method of substitution for evaluating indefinite integrals. The key strategy involves identifying a substitution $u = g(x)$ such that $g'(x)$ (or a constant multiple) appears in the integrand, transforming the integral into a standard form. Special attention is given to: - Trigonometric substitutions for quadratic forms ($\tan^{-1}$ for $x^2+a^2$, $\sin^{-1}$ for $\sqrt{a^2-x^2}$) - Algebraic substitutions for composite power functions and rational functions - Completing the square to reduce general quadratics to standard forms - Recognizing the pattern $\int \frac{f'(x)}{f(x)}dx = \ln|f(x)|$ for logarithmic integration