Lesson 1-2: Integration Techniques - Indefinite & Definite Integrals

Integration Techniques - Indefinite & Definite Integrals

Master fundamental integration techniques including indefinite and definite integrals, substitution methods, integration by parts, and geometric interpretations. Learn to apply these techniques to solve complex calculus problems and real-world applications.

Learning Objectives

Indefinite Integrals

Master antiderivatives and basic integration rules

Definite Integrals

Apply the Fundamental Theorem of Calculus and geometric interpretations

Integration Techniques

Master substitution and integration by parts methods

Real-World Applications

Apply integration to physics, economics, and engineering problems

Core Knowledge Points

Indefinite Integrals & Basic Rules

An antiderivative $F$ of a function $f$ satisfies $F'(x) = f(x)$ . The indefinite integral is the general antiderivative plus a constant:

$\int f(x) \, dx = F(x) + C$

where $C$ is the constant of integration

Basic Integration Rules:

• $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$ )
• $\int e^x \, dx = e^x + C$
• $\int \frac{1}{x} \, dx = \ln|x| + C$
• $\int \sin x \, dx = -\cos x + C$
• $\int \cos x \, dx = \sin x + C$

Definite Integrals & Fundamental Theorem

The definite integral represents the signed area under the curve and is computed using the Fundamental Theorem of Calculus:

$\int_a^b f(x) \, dx = F(b) - F(a)$

where $F$ is any antiderivative of $f$

Geometric Interpretation:

• Positive area above the x-axis
• Negative area below the x-axis
• Net area = sum of positive and negative areas

Integration Techniques

Substitution (u-substitution)

$\int f(g(x))g'(x) \, dx = \int f(u) \, du$

Let u = g(x), then du = g'(x)dx

Integration by Parts

$\int u \, dv = uv - \int v \, du$

Choose u and dv strategically

Real-World Applications

Physics and Engineering Applications

Work and Energy

Work done by a variable force: $W = \int_a^b F(x) \, dx$

Calculate work done by forces that vary with position

Area and Volume

Area between curves: $A = \int_a^b |f(x) - g(x)| \, dx$

Volume of revolution: $V = \pi \int_a^b [f(x)]^2 \, dx$

Comprehensive Example Analysis

Substitution Method Example

Find: $\int \frac{\ln x}{x} \, dx$

1. Identify u and du

Let $u = \ln x$ , then $du = \frac{1}{x} \, dx$

2. Substitute

$\int \frac{\ln x}{x} \, dx = \int u \, du$

3. Integrate

$\int u \, du = \frac{u^2}{2} + C = \frac{(\ln x)^2}{2} + C$

Integration by Parts Example

Find: $\int x e^x \, dx$

1. Choose u and dv

Let $u = x$ , $dv = e^x \, dx$

Then $du = dx$ , $v = e^x$

2. Apply Integration by Parts

$\int x e^x \, dx = x e^x - \int e^x \, dx$

3. Complete the Integration

$\int x e^x \, dx = x e^x - e^x + C = e^x(x - 1) + C$

Practice Problems

Problem 1: Basic Integration

Evaluate the following indefinite integrals:

a) $\int (3x^2 + 2x - 5) \, dx$

b) $\int \frac{1}{x^3} \, dx$

c) $\int \sin(2x) \, dx$

d) $\int e^{3x} \, dx$

Problem 2: Substitution Method

Use substitution to evaluate:

a) $\int x \sqrt{x^2 + 1} \, dx$

b) $\int \frac{x}{x^2 + 4} \, dx$

c) $\int \cos^3 x \sin x \, dx$

Problem 3: Definite Integrals

Evaluate the following definite integrals:

a) $\int_0^2 (x^2 + 3x) \, dx$

b) $\int_1^e \frac{\ln x}{x} \, dx$

c) $\int_0^{\pi/2} \sin x \, dx$

Key Takeaways

Indefinite Integrals

Find antiderivatives using basic rules and always include the constant of integration

Fundamental Theorem

Use F(b) - F(a) to evaluate definite integrals and interpret as signed area

Integration Techniques

Master substitution and integration by parts for complex integrals

Applications

Apply integration to calculate areas, volumes, work, and other physical quantities

Advanced Insights

Integration by Parts Strategy: Choose u as the function that becomes simpler when differentiated, and dv as the function that's easy to integrate.

Substitution Strategy: Look for composite functions where the derivative of the inner function appears as a factor.

Definite Integral Properties: $\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx$ and $\int_a^b f(x) \, dx = \int_a^c f(x) \, dx + \int_c^b f(x) \, dx$ .

Common Pitfalls

• Forgetting the constant of integration in indefinite integrals
• Not changing the limits of integration when using substitution in definite integrals
• Choosing u and dv incorrectly in integration by parts
• Confusing definite and indefinite integrals in problem-solving

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Integration Techniques - Indefinite & Definite Integrals

Learning Objectives

Indefinite Integrals

Definite Integrals

Integration Techniques

Real-World Applications

Core Knowledge Points

Indefinite Integrals & Basic Rules

Definite Integrals & Fundamental Theorem

Integration Techniques

Substitution (u-substitution)

Integration by Parts

Real-World Applications

Physics and Engineering Applications

Work and Energy

Area and Volume

Comprehensive Example Analysis

Substitution Method Example

Find: ∫ln⁡xx dx\int \frac{\ln x}{x} \, dx∫xlnx​dx

1. Identify u and du

2. Substitute

3. Integrate

Integration by Parts Example

Find: ∫xex dx\int x e^x \, dx∫xexdx

1. Choose u and dv

2. Apply Integration by Parts

3. Complete the Integration

Practice Problems

Problem 1: Basic Integration

Problem 2: Substitution Method

Problem 3: Definite Integrals

Key Takeaways

Indefinite Integrals

Fundamental Theorem

Integration Techniques

Applications

Advanced Insights

Common Pitfalls

Find: $\int \frac{\ln x}{x} \, dx$

Find: $\int x e^x \, dx$