--- title: Integration description: Integration is the process of finding the area under a curve or accumulating quantities. You will understand how to calculate definite and indefinite integrals, apply integration techniques to find volumes and arc lengths, and solve fundamental accumulation problems. category: mathematics subcategory: calculus difficulty: beginner, intermediate, advanced url: /subject/integration --- # Integration Integration is the process of finding the area under a curve or accumulating quantities. You will understand how to calculate definite and indefinite integrals, apply integration techniques to find volumes and arc lengths, and solve fundamental accumulation problems. ## Where to start Start with Khan Academy Integration, a free unit that mixes short videos with graded practice on Riemann sums, the fundamental theorem, substitution and integration by parts. If you only use one resource, make it David Jerison's Single Variable Calculus (MIT 18.01SC) from MIT OpenCourseWare, a self-paced course with problem sets and exams with solutions. Professor Leonard's Calculus II is the natural next step for techniques and applications. ## Available Resources 1 Books • 7 Courses • 5 Websites ## Courses ### 1. Real Analysis (MIT 18.100A) **Author:** Casey Rodriguez Rigorous foundations of calculus: real numbers, convergence of sequences and series, continuity, differentiability, the Riemann integral, sequences and series of functions, uniform convergence and interchange of limits. Includes 25 lecture videos, notes, problem sets and exams, and teaches how to read and construct proofs. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-100a-real-analysis-fall-2020/ **Tags:** real-analysis, proofs, sequences-and-series, uniform-convergence, riemann-integral ### 2. Essence of Calculus Master integration with Grant Sanderson! This calculus course unlocks the essence of integration through intuitive visuals. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr **Tags:** calculus, integration, visual-intuition, 3blue1brown, derivatives ### 3. Calculus II Master integral calculus with Professor Leonard's comprehensive Calculus II course. Learn techniques, applications & problem-solving skills! **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLDesaqWTN6EQ2J4vgsN1HyBeRADEh4Cw- **Tags:** calculus, integration-techniques, lecture-series, sequences-series, worked-examples ### 4. Calculus Open Textbook (MIT RES.18-001) **Author:** Gilbert Strang The full open textbook, 3rd edition, covering single-variable and multivariable calculus in depth: derivatives, integrals, exponentials, infinite series, vectors, partial derivatives, multiple integrals and vector calculus, with many applications. An accompanying study guide and instructor's manual support self-study or teaching a complete sequence. **Difficulty:** Beginner | **Price:** Free **Link:** https://ocw.mit.edu/courses/res-18-001-calculus-fall-2023/ **Tags:** calculus, derivatives, integrals, infinite-series, multivariable-calculus, vector-calculus ### 5. Single Variable Calculus (MIT 18.01SC) **Author:** David Jerison MIT's self-paced 18.01SC covers differentiation, applications of derivatives, integration, techniques of integration, and infinite series, with problem sets, recitation videos, and exams with solutions. Completing it gives the standard first-semester calculus foundation. **Difficulty:** Beginner | **Price:** Free **Link:** https://ocw.mit.edu/courses/18-01sc-single-variable-calculus-fall-2010/ **Tags:** single-variable-calculus, derivatives, integrals, infinite-series ### 6. Calculus 1: Integration Robert Ghrist's University of Pennsylvania course on antidifferentiation, covering substitution, integration by parts, trigonometric substitution, partial fractions, improper integrals and the fundamental theorem. Its Taylor-series-first approach assumes you finished the earlier parts of his single-variable sequence. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://www.coursera.org/learn/integration-calculus **Tags:** courses, mathematics-statistics, calculus ### 7. Introduction to Complex Analysis **Author:** Petra Bonfert-Taylor Wesleyan University course on complex analysis covering complex numbers, analytic functions, conformal mappings, Möbius transformations, contour integration, Cauchy's theorem, power and Laurent series, and residues. Learners finish able to evaluate contour integrals and analyze complex-differentiable functions. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://www.coursera.org/learn/complex-analysis **Tags:** complex-analysis, analytic-functions, conformal-mapping, contour-integration, residue-theorem ## Websites ### 1. Khan Academy Integration Khan Academy's AP Calculus BC unit on integration and accumulation of change, mixing short videos with graded practice. Covers Riemann sums, the fundamental theorem, u-substitution, integration by parts, partial fractions, and improper integrals, with immediate feedback on each attempt. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/ap-calculus-bc/bc-integration-new **Tags:** calculus, integration, riemann-sums, ap-calculus, practice-problems ### 2. Integral Table B.E. Shapiro's Creative Commons reference table of integrals, offered as a full PDF, a one-page condensed sheet, and LaTeX source. Use it to look up standard antiderivative forms while working problems rather than to learn technique. **Difficulty:** Beginner | **Price:** Free **Link:** https://integral-table.com/ **Tags:** integration, reference-table, antiderivatives, calculus, formula-sheet ### 3. Paul's Online Math Notes - Calculus II Paul Dawkins' Lamar University course notes covering the second-semester calculus sequence: substitution, integration by parts, trig substitution, partial fractions, improper integrals, applications, sequences and series. Each section pairs worked examples with practice problems and full solutions. **Difficulty:** Beginner | **Price:** Free **Link:** https://tutorial.math.lamar.edu/classes/calcii/calcii.aspx **Tags:** calculus, integration-techniques, sequences-series, lecture-notes, practice-problems ### 4. integral-calculator.com Integral-Calculator.com is an online calculator for integrals that computes indefinite and definite integrals and provides step-by-step solutions to illustrate integration techniques, making it a practical learning aid for calculus students. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://integral-calculator.com **Tags:** websites, mathematics-statistics, calculus ### 5. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Books ### 1. A First Course in Calculus **Author:** Serge Lang Serge Lang's brief original edition of his first-year calculus text, reissued by Springer. It treats limits, derivatives, and the integral with short proofs and minimal apparatus, suited to readers who want the standard sequence without a thousand-page textbook. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0387953272?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, calculus --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/integration