Using Integration Tables
Textbook Reference
| Primary source | OpenStax Calculus Volume 2, Appendix: “Table of Integrals” |
| Direct link | https://openstax.org/books/calculus-volume-2/pages/b-table-of-integrals |
| Textbook used in class | Stewart, Calculus, Section 7.5 and Reference Pages |
Opening Scenario
Even with all the techniques from Chapter 7, some integrals resist elementary methods or produce lengthy computations. Integration tables collect hundreds of standard antiderivative formulas. Reading a table entry correctly -- matching the form, identifying the parameters, and applying the formula -- is a practical skill. Reduction formulas in tables also allow complex integrals to be systematically reduced to simpler ones.
Quick Reference
How to use an integration table:
- Identify the form of the integrand. Match it to a table entry by comparing structure (e.g., involves $\sqrt{a^2-x^2}$, or is of the form $x^n e^{ax}$).
- Read off the parameters ($a$, $b$, $n$, etc.) from the integrand.
- Substitute those parameters into the table formula.
- Simplify if needed.
Reduction formulas express $\int f^n(x)\,dx$ in terms of $\int f^{n-2}(x)\,dx$ (or some other lower power). Apply repeatedly until reaching a base case.
Key Concepts
1. Matching Forms
Tables organize entries by the type of expression in the integrand. Common categories:
- Rational functions (involving $ax + b$ or $ax^2 + bx + c$)
- Expressions with $\sqrt{a^2 - x^2}$, $\sqrt{a^2 + x^2}$, $\sqrt{x^2 - a^2}$
- Trigonometric integrals ($\sin^n$, $\cos^n$, $\tan^n$, products)
- Exponential and logarithm combinations
- Inverse trigonometric forms
The key is recognizing which category the integrand falls into before searching the table.
2. Adjusting the Form
Sometimes the integrand nearly matches a table entry but differs by a constant factor or requires a substitution to match. For example, a table might have $\int\sqrt{a^2-u^2}\,du$, but your integral has $\int\sqrt{9-4x^2}\,dx$. Substitute $u = 2x$, $du = 2\,dx$ to reduce to the table form with $a = 3$.
3. Reduction Formulas
A reduction formula is a recurrence relation for an integral. For example, the table might list:
$$\int\sin^n x\,dx = -\frac{\sin^{n-1}x\cos x}{n} + \frac{n-1}{n}\int\sin^{n-2}x\,dx.$$
To evaluate $\int\sin^5 x\,dx$: apply the formula once to get a multiple of $\int\sin^3 x\,dx$, apply again to get a multiple of $\int\sin x\,dx$, then evaluate directly.
“A table gives the answer directly without any work.” Using a table still requires matching the form, identifying parameters, and potentially doing a preliminary substitution to put the integrand in the right form. The table replaces the derivation of the formula, not the thinking about how to apply it.
Worked Example
Use a table to evaluate $\displaystyle\int\sqrt{x^2+9}\,dx$.
A table of integrals includes the entry: $$\int\sqrt{u^2+a^2}\,du = \frac{u}{2}\sqrt{u^2+a^2} + \frac{a^2}{2}\ln\!\left(u + \sqrt{u^2+a^2}\right) + C.$$
Match: $u = x$, $a^2 = 9$, so $a = 3$.
Apply: $$\int\sqrt{x^2+9}\,dx = \frac{x}{2}\sqrt{x^2+9} + \frac{9}{2}\ln\!\left(x + \sqrt{x^2+9}\right) + C.$$
Compare with trig substitution: The same result can be derived from $x = 3\tan\theta$ and $\int\sec^3\theta\,d\theta$, confirming the table entry is correct and showing that the table saves significant work.
Common Errors Summary
| Error | Example | Correction |
|---|---|---|
| Using the wrong table form | Applying the $\sqrt{a^2-u^2}$ formula to $\sqrt{u^2+a^2}$ | Match the sign inside the square root to the correct table entry |
| Forgetting to adjust for a change of variable | Applying $\int\sqrt{a^2-u^2}\,du$ directly to $\int\sqrt{9-4x^2}\,dx$ without substituting $u=2x$ | Substitute first to match the exact form; then adjust by dividing by $du/dx$ |
| Treating a reduction formula as a closed-form answer | Stopping after one application of a reduction formula | A reduction formula must be applied until the base case; then back-substitute each time |
Leveled Practice
Level 1 -- Identify the Form
Problem 1. For each integral, identify which category of table entry to look up.
(a) $\displaystyle\int\frac{1}{x\sqrt{4-x^2}}\,dx$ (b) $\displaystyle\int x^3 e^{2x}\,dx$ (c) $\displaystyle\int\cos^6 x\,dx$
Show answer
(a) Involves $\sqrt{a^2-x^2}$ with $a=2$; look in the “expressions with $\sqrt{a^2-x^2}$” section.
(b) Product of polynomial and exponential; look for $\int x^n e^{ax}\,dx$ or use the reduction formula for this type.
(c) Power of cosine; look for a reduction formula $\int\cos^n x\,dx$ or use the half-angle identity repeatedly.
Mastery Checklist
Mental Model
An integration table is a library of pre-computed derivations. The skill is knowing which shelf to look on (what form the integrand matches) and how to read the entry (what the parameters mean). The table does not think for you; it just saves you from re-deriving formulas that have already been established.
Connections
Looking back
- Integration strategy (Section 7.5): A table is the appropriate tool after the main techniques have been tried and when the derivation would be lengthy.
- Trig substitution (Section 7.3): Many table entries for expressions with square roots are derived via trig substitution.
Looking ahead
- Numerical integration (Section 7.7): When no table entry fits and no closed form exists, numerical methods take over.
Back to Techniques of Integration | Next: Numerical Integration