Integration Cheat Sheet

This cheat sheet covers the fundamental concepts of integration, including definite and indefinite integrals, their properties, the Fundamental Theorem of Calculus, and various integration techniques like substitution and partial fractions.

Core Principles

• The definite integral represents the net area under a curve between two points.
• The indefinite integral represents a family of functions (antiderivatives).
• The Fundamental Theorem of Calculus links differentiation and integration.
• Integration by substitution is used to simplify integrals by changing variables.
• Partial fractions decompose complex rational functions into simpler, integrable parts.

Action Steps

• Identify the type of integral (definite or indefinite).
• Choose an appropriate integration technique (e.g., substitution, partial fractions).
• Find the antiderivative of the integrand.
• For definite integrals, evaluate the antiderivative at the limits of integration and subtract.
• For indefinite integrals, add the constant of integration 'C'.

Formulas

• $ \int_{a}^{b} f(x) dx = F(b) - F(a) $
• $ \int f(x) dx = F(x) + C $
• $ \int_{a}^{b} f(x) dx = - \int_{b}^{a} f(x) dx $
• $ \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx = \int_{a}^{b} f(x) dx $
• $ \int (f(x) + g(x)) dx = \int f(x) dx + \int g(x) dx $
• $ \int c f(x) dx = c \int f(x) dx $
• $ \int x^n dx = \frac{x^{n+1}}{n+1} + C \text{ (for } n \neq -1) $
• $ \int \frac{1}{x} dx = \ln|x| + C $
• $ \int e^x dx = e^x + C $
• $ \int \cos x dx = \sin x + C $
• $ \int \sin x dx = -\cos x + C $

Key Terms

• Definite Integral: An integral with upper and lower limits of integration, representing a numerical value (often area).
• Indefinite Integral: An integral without limits, representing a family of functions (antiderivatives) plus a constant C.
• Antiderivative: A function whose derivative is the original function.
• Riemann Sum: An approximation of a definite integral using a sum of areas of rectangles.
• Fundamental Theorem of Calculus (FTC): Connects definite integrals and antiderivatives, stating that the integral of a rate of change is the total change.
• Integration by Substitution: A technique to simplify integrals by replacing a part of the integrand with a new variable (u).
• Partial Fractions: A method to decompose a rational function into a sum of simpler rational functions for easier integration.

Timeline

• 11/5/26: MTH1020 Seminar: Introduction to Integration.
• Past: Previously covered: Applications of Differentiation.
• Today: Begin new topic: Integration. Focus on definite integrals and their interpretation as area.
• Upcoming: Rolyn Arianrhod talk on Wed at 2pm in MLC - G39, 9 Rainforest Walk.

People

• Daniel Mathews: Instructor/Lecturer
• Rolyn Arianrhod: Guest Speaker (Talk)