Using Technology for Integration
Textbook Reference
| Primary source | OpenStax Calculus Volume 2, Section 3.5: “Other Strategies for Integration” |
| Direct link | https://openstax.org/books/calculus-volume-2/pages/3-5-other-strategies-for-integration |
| Textbook used in class | Stewart, Calculus, Section 7.6: “Integration Using Tables and Computer Algebra Systems” |
Opening Scenario
Computer algebra systems (CAS) -- including Wolfram Alpha, Maple, Mathematica, and the integration tools in many graphing calculators -- can evaluate many integrals symbolically. They do not replace understanding; they change the nature of the skill. Instead of computing, the skill becomes interpreting the output, verifying it, and recognizing when a CAS answer is equivalent to what you expect even though it looks different.
Quick Reference
How to use a CAS for integration:
- Enter the integrand in standard notation (check syntax carefully).
- Read the output, including any constants or assumptions.
- Verify: differentiate the output and confirm you get the original integrand.
- Compare forms: two antiderivatives that look different may be equal (they differ by a constant).
Key Concepts
1. Equivalent but Different-Looking Answers
A CAS may return an antiderivative in a form that looks different from the one you computed by hand. Both are correct if they differ by a constant.
Example. The antiderivative of $\sin x\cos x$ can be written as:
- $\dfrac{\sin^2 x}{2} + C$ (from substitution $u = \sin x$)
- $-\dfrac{\cos^2 x}{2} + C$ (from substitution $u = \cos x$)
- $-\dfrac{\cos(2x)}{4} + C$ (from the double-angle formula)
All three are correct: they differ by the constants $\frac{1}{2}$ and $0$. To check equivalence, differentiate both answers and confirm they equal the original integrand, or subtract them and verify the result is a constant.
2. Verifying a CAS Answer
The most reliable check: differentiate the CAS output and confirm it matches the integrand.
Example. A CAS returns $\sqrt{x^2-1} - \arctan\!\left(\dfrac{1}{\sqrt{x^2-1}}\right) + C$ for some integral. Differentiate to verify:
$\dfrac{d}{dx}\left[\sqrt{x^2-1} - \arctan\!\left(\dfrac{1}{\sqrt{x^2-1}}\right)\right] = \dfrac{x}{\sqrt{x^2-1}} - \dfrac{1}{1+\frac{1}{x^2-1}}\cdot\dfrac{-x}{(x^2-1)^{3/2}}$
$= \dfrac{x}{\sqrt{x^2-1}} + \dfrac{x(x^2-1)}{x^2\cdot(x^2-1)} \cdot \dfrac{1}{\sqrt{x^2-1}} = \dfrac{x}{\sqrt{x^2-1}} + \dfrac{1}{x\sqrt{x^2-1}}$.
This matches the original integrand if the original was $\dfrac{x}{\sqrt{x^2-1}} + \dfrac{1}{x\sqrt{x^2-1}}$. If not, something went wrong with the CAS input.
3. When a CAS Returns No Answer
A CAS may report that an integral has no closed form. This does not mean the integral does not exist -- it means no elementary function has that derivative. Examples: $e^{-x^2}$, $\dfrac{\sin x}{x}$, $\sqrt{1-k^2\sin^2 x}$. These are integrated numerically.
4. The Role of Understanding
A CAS is a computation tool, not a teacher. It cannot tell you why an integral arises, what it represents geometrically, whether your setup was correct, or how to interpret the result in context. The conceptual work -- setting up the integral correctly, choosing integration bounds, interpreting units -- requires the understanding built through Chapters 4--7.
“If the CAS gives a different answer than I computed, my answer is wrong.” Not necessarily. If differentiating both answers gives the same integrand, both answers are correct antiderivatives. Check by subtracting: if $F_1(x) - F_2(x)$ is a constant, they are both correct.
Worked Example
Two students evaluate $\displaystyle\int\frac{2x}{x^2+1}\,dx$. Student A gets $\ln(x^2+1) + C$. A CAS returns $\ln(x^2+1)$. Are these the same?
Yes. The CAS omits the explicit $+C$ (it is understood). Both answers are correct.
A CAS returns $\text{arcsec}|x| + C$ for an integral. Your by-hand answer was $\arctan\!\left(\sqrt{x^2-1}\right) + C$. Are these equal?
Check by differentiation, or use the identity $\text{arcsec}|x| = \arctan\!\left(\sqrt{x^2-1}\right)$ for $x > 1$. Yes, they are equal (and differ by 0).
Common Errors Summary
| Error | Example | Correction |
|---|---|---|
| Accepting a CAS answer without verifying | Taking $F(x)$ at face value without differentiating | Differentiate $F(x)$ and confirm $F'(x) = f(x)$; CAS systems have bugs and input errors are common |
| Concluding two different forms are both wrong | Seeing $-\frac{\cos^2 x}{2}$ and $\frac{\sin^2 x}{2}$ and thinking one must be wrong | Subtract to get $\frac{\sin^2 x + \cos^2 x}{2} = \frac{1}{2}$, a constant; both are correct |
| Mistyping the integrand into the CAS | Entering $x/sqrt(x^2-1)$ when the integrand was $x/sqrt(x^2+1)$ | Double-check the CAS input before using the output; a sign error in the input gives a wrong formula |
Leveled Practice
Level 1 -- Verification
Problem 1. A CAS returns $\dfrac{x^3}{3}\ln x - \dfrac{x^3}{9} + C$ for $\displaystyle\int x^2\ln x\,dx$. Verify by differentiating.
Show answer
$\dfrac{d}{dx}\!\left[\dfrac{x^3}{3}\ln x - \dfrac{x^3}{9}\right] = \dfrac{x^2}{1}\cdot\ln x + \dfrac{x^3}{3}\cdot\dfrac{1}{x} - \dfrac{x^2}{3} = x^2\ln x + \dfrac{x^2}{3} - \dfrac{x^2}{3} = x^2\ln x$.
Confirmed.
Level 2 -- Equivalence Check
Problem 2. Are $\ln|\tan x| + C$ and $\ln|\sin x| - \ln|\cos x| + C$ the same antiderivative?
Show answer
$\ln|\tan x| = \ln\!\left|\dfrac{\sin x}{\cos x}\right| = \ln|\sin x| - \ln|\cos x|$.
Yes, they are identical (not just equal up to a constant -- exactly equal).
Mastery Checklist
Mental Model
A CAS is a high-powered reference book that can look up integrals much faster than any table and handle far more complex cases. Using it well means knowing how to frame the question, read the answer, and verify the output. It does not know whether you set up the problem correctly, whether the limits of integration match the physical situation, or what the answer means. Those questions belong to you.
Connections
Looking back
- Integration strategy (Section 7.5): A CAS is a last resort after known techniques have been tried, or a quick check after manual computation.
Looking ahead
- Numerical integration (Section 7.7): When a CAS returns no closed form, numerical methods provide approximate answers.
Back to Techniques of Integration | Next: Numerical Integration