Integration using tables is an efficient technique for solving complex integrals in mathematics. This method leverages pre-established integrals from a reference table, simplifying the process of integration by matching given problems to known solutions. By familiarizing yourself with the most common entries in these tables, you can significantly enhance your ability to solve integrals quickly and accurately.
Using standard table integrals means using a list of known antiderivatives to quickly solve calculus problems. The key ideas are pattern recognition, substitution, and simplification.
How Standard Table Integrals Work
- Pattern matching: You look at a complex function and compare it to a list of known formulas to find a match.
- Algebraic adjustment: You change the look of your problem using math rules so it fits a standard formula on the table.
- Substitution (u-substitution): You replace a part of your function with a new variable like u to make it match a table entry.
Why We Use Them
- Saves time: You skip long, hard steps like trigonometric substitution or partial fractions.
- Reduces errors: Standard formulas are already checked and proven to be true.
- Solves hard forms: They help find answers for tough functions with square roots or trigonometric parts.
The References section below contains additional standard tables for your use.
| Common Functions | Function | Integral |
| Constant | ∫ a dx | ax + C |
| Variable | ∫ x dx | x2/2 + C |
| Square | ∫ x2 dx | x3/3 + C |
| Reciprocal | ∫ (1/x) dx | ln|x| + C |
| Exponential | ∫ ex dx | ex + C |
| ∫ ax dx | ax/ln(a) + C | |
| ∫ ln(x) dx | x ln(x) − x + C | |
| Trigonometry (x in radians) | ∫ cos(x) dx | sin(x) + C |
| ∫ sin(x) dx | -cos(x) + C | |
| ∫ sec2(x) dx | tan(x) + C | |
| Rules | Function | Integral |
| Multiplication by constant | ∫ c f(x) dx | c ∫ f(x) dx |
| Power Rule (n≠−1) | ∫ xn dx | (xn+1)/(n+1) + C |
| Sum Rule | ∫ (f + g) dx | ∫ f dx + ∫ g dx |
| Difference Rule | ∫ (f – g) dx | ∫ f dx – ∫g dx |
| Integration by Parts | See Integration by Parts | |
| Substitution Rule | See Integration by Substitution |
References
“Calculus Cheat Sheet”. 2022. tutorial.math.lamar.edu. https://tutorial.math.lamar.edu/pdf/Calculus_Cheat_Sheet_All.pdf.
“Integration Cheat Sheet.” PMTEduction, Accessed July 15, 2026. https://pmt.physicsandmathstutor.com/download/Maths/A-level/Pure/Integration-2/Cheat-Sheets-Edexcel/Ch.11 Integration.pdf.
[ ℰ ] Strang, Gilbert, and Edwin “Jed” Herman. “A Table of Integrals – Calculus Volume 1.” OpenStax, March 30, 2016. https://openstax.org/books/calculus-volume-1/pages/a-table-of-integrals.
[ ℰ ] “What Is the Method of Integration Using Tables?” StudySmarter UK, Accessed July 15, 2026. https://www.studysmarter.co.uk/explanations/math/calculus/integration-using-tables/.
[ ℰ ] This exceptional reference is highly recommended for your consideration.
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