Calculus: Integration Techniques and Analytical Geometry Study Guide

Fundamental Principles of Calculus

• Definition of Integration: Integration is defined as the reverse process of differentiation.
• The Integration Process:
  • Given a function $y = f(x)$.
  • Differentiation: $\frac{dy}{dx} = f'(x)$.
  • Integration: $\int f'(x) dx = f(x) + C$.
• Notation:
  • The symbol $\int$ is the integration sign.
  • The function being integrated, $f(x)$, is called the integrand.
  • $C$ represents the constant of integration.
• Differential Relation: The differential $dy$ is equal to $f'(x) dx$.

Basic Rules and Formulas

• Linearity of Integration:
  • $\int [f(x) \pm g(x)] dx = \int f(x) dx \pm \int g(x) dx$
  • $\int k f(x) dx = k \int f(x) dx$ (where $k$ is a constant).
• Power Rule for Integration:
  • $\int x^n dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$).
  • General form: $\int [f(x)]^n f'(x) dx = \frac{[f(x)]^{n+1}}{n+1} + C$.
• Integration of a Constant:
  • $\frac{d}{dx}(x) = 1 \rightarrow \int 1 dx = x + C$.
• Exponential Functions:
  • $\int e^x dx = e^x + C$.
  • $\int e^{ax} dx = \frac{e^{ax}}{a} + C$.
  • $\int a^x dx = \frac{a^x}{\ln(a)} + C$.
• Logarithmic Rule:
  • If the derivative of the denominator is present in the numerator ($\text{niche function, uper derivative}$), the integral is the natural log of the denominator:
  • $\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + C$.
  • Example: $\int \frac{e^x}{e^x + 3} dx = \ln|e^x + 3| + C$.
  • Example: $\int \tan(x) dx = \int \frac{\sin(x)}{\cos(x)} dx = -\ln|\cos(x)| + C = \ln|\sec(x)| + C$.
  • Example: $\int \cot(x) dx = \int \frac{\cos(x)}{\sin(x)} dx = \ln|\sin(x)| + C$.

Integration by Parts (IBP)

• Formula: $\int u \cdot v dx = u \int v dx - \int (u' \cdot \int v dx) dx$.
• Selection Priority (ILATE Rule): To choose the first function ($u$), follow this hierarchy:
  1. I: Inverse Trigonometric functions
  2. L: Logarithmic functions
  3. A: Algebraic functions
  4. T: Trigonometric functions
  5. E: Exponential functions
• Special Note: Inverse and logarithmic functions are always prioritized as the first function ($u$).

Trigonometric Integration and Identities

• Fundamental Integrals:
  • $\int \sin(x) dx = -\cos(x) + C$
  • $\int \cos(x) dx = \sin(x) + C$
• Square Identities for Integration:
  • $\sin^2(x) = \frac{1 - \cos(2x)}{2}$
  • $\cos^2(x) = \frac{1 + \cos(2x)}{2}$
• Evaluation of Square Trigonometric Integrals:
  • $\int \sin^2(x) dx = \int \frac{1 - \cos(2x)}{2} dx = \frac{1}{2} \int (1 - \cos(2x)) dx = \frac{1}{2} (x - \frac{\sin(2x)}{2}) + C = \frac{x - \sin(x)\cos(x)}{2} + C$
  • $\int \cos^2(x) dx = \int \frac{1 + \cos(2x)}{2} dx = \frac{1}{2} (x + \frac{\sin(2x)}{2}) + C = \frac{x + \sin(x)\cos(x)}{2} + C$
• Product-to-Sum Formulas:
  • $2 \sin(\alpha) \cos(\beta) = \sin(\alpha + \beta) + \sin(\alpha - \beta)$
  • $2 \cos(\alpha) \sin(\beta) = \sin(\alpha + \beta) - \sin(\alpha - \beta)$
  • $2 \cos(\alpha) \cos(\beta) = \cos(\alpha + \beta) + \cos(\alpha - \beta)$
  • $2 \sin(\alpha) \sin(\beta) = \cos(\alpha - \beta) - \cos(\alpha + \beta)$
• Addition and Subtraction Formulas:
  • $\sin(\alpha \pm \beta) = \sin(\alpha)\cos(\beta) \pm \cos(\alpha)\sin(\beta)$
  • $\cos(\alpha \pm \beta) = \cos(\alpha)\cos(\beta) \mp \sin(\alpha)\sin(\beta)$
  • $\tan(\alpha \pm \beta) = \frac{\tan(\alpha) \pm \tan(\beta)}{1 \mp \tan(\alpha)\tan(\beta)}$

Standard Substitutions for Radicals

• For $\sqrt{a^2 - x^2}$, substitute $x = a \sin(\theta)$.
• For $\sqrt{a^2 + x^2}$, substitute $x = a \tan(\theta)$.
• For $\sqrt{x^2 - a^2}$, substitute $x = a \sec(\theta)$.

Partial Fractions Methodology

Integration by partial fractions is used when the integrand is a rational function. The technique depends on the nature of the denominator $Q(x)$:

• Case I: $Q(x)$ has non-repeated linear factors.
• Case II: $Q(x)$ has repeated linear factors.
• Case III: $Q(x)$ has non-repeated irreducible quadratic factors.
• Case IV: $Q(x)$ has repeated irreducible quadratic factors.

Detailed Worked Examples

Example 1: Integration by Parts with Natural Log

• Evaluate: $I = \int \ln(x) dx$
• Solution: Let $u = \ln(x)$ and $dv = 1 dx$. Then $du = \frac{1}{x} dx$ and $v = x$.
• Using IBP: $I = x \ln(x) - \int x \cdot \frac{1}{x} dx$
• $I = x \ln(x) - \int 1 dx = x \ln(x) - x + C$.

Example 2: Repeated Integration by Parts

• Evaluate: $I = \int x^2 e^{5x} dx$
• Solution:
  • 1st iteration ($u = x^2, dv = e^{5x} dx$): $I = x^2 \frac{e^{5x}}{5} - \int \frac{e^{5x}}{5} (2x) dx = \frac{x^2 e^{5x}}{5} - \frac{2}{5} \int x e^{5x} dx$
  • 2nd iteration ($u = x, dv = e^{5x} dx$): $\int x e^{5x} dx = x \frac{e^{5x}}{5} - \int \frac{e^{5x}}{5} dx = \frac{x e^{5x}}{5} - \frac{e^{5x}}{25}$
  • Final Assembly: $I = \frac{x^2 e^{5x}}{5} - \frac{2}{5} (\frac{x e^{5x}}{5} - \frac{e^{5x}}{25}) + C = \frac{x^2 e^{5x}}{5} - \frac{2x e^{5x}}{25} + \frac{2 e^{5x}}{125} + C$.

**Example 3: Integral of $\sqrt{a^2 - x^2}$

• Evaluate: $I = \int \sqrt{a^2 - x^2} dx$
• Solution: Let $u = \sqrt{a^2 - x^2}$ and $dv = 1 dx$.
• $I = x\sqrt{a^2 - x^2} - \int x \frac{1}{2\sqrt{a^2 - x^2}}(-2x) dx$
• $I = x\sqrt{a^2 - x^2} + \int \frac{x^2}{\sqrt{a^2 - x^2}} dx$
• Using the trick $x^2 = -(a^2 - x^2) + a^2$:
• $I = x\sqrt{a^2 - x^2} - \int \frac{a^2 - x^2}{\sqrt{a^2 - x^2}} dx + \int \frac{a^2}{\sqrt{a^2 - x^2}} dx$
• $I = x\sqrt{a^2 - x^2} - I + a^2 \sin^{-1}(\frac{x}{a})$
• $2I = x\sqrt{a^2 - x^2} + a^2 \sin^{-1}(\frac{x}{a})$
• $I = \frac{x\sqrt{a^2 - x^2}}{2} + \frac{a^2}{2} \sin^{-1}(\frac{x}{a}) + C$.

Example 4: Completing the Square

• Evaluate: $I = \int \frac{1}{x^2 - 4x + 5} dx$
• Solution: Complete the square for $x^2 - 4x + 5$.
• $x^2 - 4x + 4 + 1 = (x - 2)^2 + 1^2$
• $I = \int \frac{1}{(x - 2)^2 + 1} dx = \tan^{-1}(x - 2) + C$.

Example 5: Multi-term Trigonometric Integration

• Evaluate: $I = \int \sin(7x) \cos(3x) dx$
• Solution: Use product-to-sum identity $\sin(\alpha)\cos(\beta) = \frac{1}{2}[\sin(\alpha + \beta) + \sin(\alpha - \beta)]$
• $I = \frac{1}{2} \int (\sin(10x) + \sin(4x)) dx$
• $I = \frac{1}{2} (-\frac{\cos(10x)}{10} - \frac{\cos(4x)}{4}) + C = -\frac{\cos(10x)}{20} - \frac{\cos(4x)}{8} + C$.

Practical Calculations and Identities

• Evaluate: $\int (\frac{2}{x} + 3e^x) dx$
  • $= 2 \int \frac{1}{x} dx + 3 \int e^x dx = 2 \ln|x| + 3e^x + C$.
• Derivative and Integral relation:
  • $\frac{d}{dx}(\sin(2x)) = 2\cos(2x)$
  • $\int \cos(2x) dx = \frac{\sin(2x)}{2} + C$.

Questions & Discussion

Question 1: Evaluate $\int \frac{x}{x^2 + 2x + 2} dx$

• Solution Strategy: This requires splitting the numerator to create the derivative of the denominator.
• Derivative of $x^2 + 2x + 2$ is $2x + 2$.
• Multiply and divide by 2: $I = \frac{1}{2} \int \frac{2x}{x^2 + 2x + 2} dx$
• Add and subtract 2: $I = \frac{1}{2} \int \frac{2x + 2 - 2}{x^2 + 2x + 2} dx$
• $I = \frac{1}{2} \int \frac{2x + 2}{x^2 + 2x + 2} dx - \int \frac{1}{x^2 + 2x + 2} dx$
• $I = \frac{1}{2} \ln|x^2 + 2x + 2| - \int \frac{1}{(x+1)^2 + 1} dx$
• $I = \frac{1}{2} \ln|x^2 + 2x + 2| - \tan^{-1}(x+1) + C$.

Question 2: Evaluate $\int \frac{1}{\sqrt{x^2 - 4}} dx$

• Solution: Use substitution $x = 2 \sec(\theta)$, $dx = 2 \sec(\theta)\tan(\theta) d\theta$.
• $I = \int \frac{2 \sec(\theta)\tan(\theta)}{\sqrt{4\sec^2(\theta) - 4}} d\theta = \int \frac{2 \sec(\theta)\tan(\theta)}{2\tan(\theta)} d\theta$
• $I = \int \sec(\theta) d\theta = \ln|\sec(\theta) + \tan(\theta)| + C$
• Since $\sec(\theta) = \frac{x}{2}$ and $\tan(\theta) = \sqrt{\sec^2(\theta) - 1} = \sqrt{\frac{x^2}{4} - 1} = \frac{\sqrt{x^2 - 4}}{2}$
• $I = \ln|\frac{x}{2} + \frac{\sqrt{x^2 - 4}}{2}| + C = \ln|x + \sqrt{x^2 - 4}| - \ln(2) + C$
• Final Answer: $I = \ln|x + \sqrt{x^2 - 4}| + C$.