# **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 of basic trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) - Standard integral formulas and their application with linear arguments ($ax+b$) - Trigonometric identities for simplification ($1+\tan^2 x = \sec^2 x$, $\cos^2 x$ half-angle, etc.) - Integration by substitution (u-substitution) for composite functions - Algebraic simplification of integrands before integration (e.g., $\csc x \tan x = \sec x$) - Linearity properties of integrals (sum rule and constant multiple rule) - Verification of integration results by differentiation --- # **Important Formulas** Below are the key formulas used in this exercise: **Standard Trigonometric Integrals:** $$\int \sin(ax) \, dx = -\frac{\cos(ax)}{a} + C$$ $$\int \cos(ax) \, dx = \frac{\sin(ax)}{a} + C$$ $$\int \sec^2(ax+b) \, dx = \frac{\tan(ax+b)}{a} + C$$ $$\int \csc^2(ax+b) \, dx = -\frac{\cot(ax+b)}{a} + C$$ $$\int \sec(ax)\tan(ax) \, dx = \frac{\sec(ax)}{a} + C$$ **Logarithmic Forms:** $$\int \tan(ax) \, dx = -\frac{1}{a}\ln|\cos(ax)| + C = \frac{1}{a}\ln|\sec(ax)| + C$$ $$\int \ cot(ax) \, dx = \frac{1}{a}\ln|\sin(ax)| + C$$ $$\int \csc(ax) \, dx = -\frac{1}{a}\ln|\csc(ax) + \cot(ax)| + C$$ **Trigonometric Identities:** $$1 + \tan^2 x = \sec^2 x \quad \Rightarrow \quad \tan^2 x = \sec^2 x - 1$$ $$1 + \cot^2 x = \csc^2 x \quad \Rightarrow \quad \cot^2 x = \csc^2 x - 1$$ $$\cos^2 x = \frac{1 + \cos(2x)}{2}$$ **Substitution Rule:** $$\int f(g(x))g'(x) \, dx = \int f(u) \, du \quad \text{where } u = g(x)$$ --- # **Summary** This exercise covers the fundamental techniques of integrating trigonometric functions and their combinations. The primary strategies involve applying standard integral formulas directly, using trigonometric identities to rewrite integrands into integrable forms (such as converting $\tan^2$ to $\sec^2-1$ or simplifying products like $\csc x \tan x$ to $\sec x$), and employing substitution for composite functions. Particular attention is required for linear transformations inside trigonometric functions ($ax+b$), which introduce scaling factors of $\frac{1}{a}$ in the result. The exercise emphasizes the importance of verifying solutions by differentiating the result to recover the original integrand, reinforcing the inverse relationship between differentiation and integration.