Integration Introduction
Introduction (High School Level)
Indefinite and Definite Integrals
Overview
In this introduction, we cover the fundamentals of integration as taught in high school mathematics. You will learn about the indefinite integral as the reverse operation of differentiation, the definite integral for computing areas, and the techniques of integration by substitution and integration by parts.
Learning Goals
- Understand the concept and basic formulas of indefinite integrals
- Understand the relationship between definite integrals and area
- Master integration by substitution and integration by parts
- Understand applications of integration (area and volume)
Table of Contents
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Chapter 1 Indefinite Integral
Antiderivatives, constant of integration, basic formulas |
Proofs |
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Chapter 2 Definite Integral
Definition of the definite integral, area, fundamental theorem |
Proofs |
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Chapter 3 Integration by Substitution
Substitution method, trigonometric substitution |
Proofs |
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Chapter 4 Integration by Parts
Integration by parts, worked examples |
Proofs |
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Chapter 5 Area Calculation
Area enclosed by curves, area between two curves |
Proofs |
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Chapter 6 Volume Calculation
Volume of solids of revolution, volume by cross-sectional area |
Proofs |
Prerequisites
- Basics of differentiation (computing derivatives)
- Basics of trigonometric, exponential, and logarithmic functions
Basic Formulas
Basic Indefinite Integrals
$$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C \quad (n \neq -1)$$ $$\int \frac{1}{x} \, dx = \ln|x| + C$$ $$\int e^x \, dx = e^x + C$$ $$\int \sin x \, dx = -\cos x + C$$ $$\int \cos x \, dx = \sin x + C$$Fundamental Theorem of Calculus
When $F'(x) = f(x)$:
$$\int_a^b f(x) \, dx = F(b) - F(a) = [F(x)]_a^b$$Integration by Parts
$$\int f(x)g'(x) \, dx = f(x)g(x) - \int f'(x)g(x) \, dx$$