# **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: - Definite integration using the Fundamental Theorem of Calculus - Integration of polynomial functions and power functions - Integration of trigonometric functions ($\sin x$, $\cos x$, $\sec^2 x$) - Integration by substitution (u-substitution) for composite functions - Integration by parts for logarithmic and inverse trigonometric functions - Partial fraction decomposition for rational functions - Integration of exponential functions with fractional exponents - Algebraic manipulation and simplification of integrands - Even function properties over symmetric intervals - Power reduction formulas for trigonometric integrals --- # **Important Formulas** Below are the key formulas used in this exercise: Fundamental Theorem of Calculus: $$\int_a^b f(x) \, dx = F(b) - F(a) \quad \text{where } F'(x) = f(x)$$ Power Rule for Integration: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$ Basic Trigonometric Integrals: $$\int \cos x \, dx = \sin x + C$$ $$\int \sec^2 x \, dx = \tan x + C$$ $$\int \sin x \, dx = -\cos x + C$$ Integration by Parts: $$\int u \, dv = uv - \int v \, du$$ Power Reduction Formula: $$\cos^2 x = \frac{1 + \cos(2x)}{2}$$ Partial Fractions (distinct linear factors): $$\frac{1}{(x+a)(x+b)} = \frac{1}{b-a}\left(\frac{1}{x+a} - \frac{1}{x+b}\right)$$ --- # **Summary** This exercise provides comprehensive practice in evaluating definite integrals using various integration techniques. The questions progress from basic polynomial and simple trigonometric integration to advanced methods requiring substitution (particularly for composite functions like $\sqrt{x}$ and $\theta + \sin\theta$), integration by parts (for $\ln x$ and $\tan^{-1} y$), and partial fraction decomposition (for rational functions in $x$ and $\cos x$). Key strategies include recognizing when algebraic simplification is needed before integrating (such as dividing polynomials or multiplying by conjugates for trigonometric rational functions), applying the Fundamental Theorem of Calculus with careful evaluation at boundary points, and utilizing even/odd function properties to simplify calculations over symmetric intervals. The exercise emphasizes the importance of selecting the appropriate technique based on the integrand's structure.