Now that Week 11 has supplied the natural logarithm and exponential function, both the limit toolkit and the integration toolkit can be substantially extended. This week does both.
12.1 Indeterminate Forms
A limit of the form lim f(x)/g(x) is an indeterminate form 0/0 or ∞/∞ when both numerator and denominator tend to 0, or both tend to ±∞. Week 2 §2.8 handled simple 0/0 cases by factoring or rationalizing; those tricks do not exist for every function (try factoring (ex−1)/x as x→0). A general method is needed.
12.2 L'Hôpital's Rule
If f(x)/g(x) is 0/0 or ∞/∞ as x→a (or x→±∞), and f, g are differentiable near a with g′ ≠ 0 there, then
— provided the limit on the right exists (or is ±∞). L'Hôpital's Rule can be applied repeatedly if the new ratio f′/g′ is again indeterminate.
L'Hôpital's Rule differentiates the numerator and denominator separately — it is not the Quotient Rule. Never write d/dx[f/g] here; compute f′(x) and g′(x) independently, then form the new ratio.
Example 12.1 — L'Hôpital's Rule, 0/0
Evaluate limx→0 (ex−1)/x.
Step 1 — Confirm the indeterminate form
(e0−1)/0 = 0/0.
Step 2 — Differentiate numerator and denominator separately
d/dx[ex−1] = ex; d/dx[x] = 1.
Step 3 — Apply Eq. 12.1
limx→0 ex/1 = e0 = 1
12.3 Indeterminate Products and Differences
Forms like 0·∞ or ∞−∞ are not directly covered by Eq. 12.1, but algebraic rewriting converts them into 0/0 or ∞/∞. For a product f(x)g(x) with f→0, g→∞, rewrite as f/(1/g) (giving 0/0) or g/(1/f) (giving ∞/∞).
Example 12.2 — An indeterminate product
Evaluate limx→0+ x ln x.
Step 1 — Recognise the form and rewrite
0·(−∞) form. Rewrite as ln x1/x, now −∞/∞.
Step 2 — Apply L'Hôpital's Rule
1/x−1/x2 = −x
Step 3 — Take the limit of the new expression
limx→0+ (−x) = 0
12.4 Indeterminate Powers
Forms 1∞, 00, ∞0 arise from y=f(x)g(x). Take the natural logarithm of both sides, ln y = g(x) ln f(x), which is now a 0·∞ product (§12.3); find its limit, then exponentiate (Week 11 §11.8) to undo the logarithm.
Example 12.3 — An indeterminate power
Evaluate limx→0+ xx.
Step 1 — Take logarithms
y=xx ⇒ ln y = x ln x.
Step 2 — This limit was found in Example 12.2
limx→0+ x ln x = 0, so limx→0+ ln y = 0.
Step 3 — Exponentiate to undo the logarithm
limx→0+ xx = e0 = 1
12.5 Basic Integration Formulas: A Consolidated Table
With the natural logarithm and exponential now available, the antiderivative table from Week 9 §9.5 and Week 11 §11.6/§11.8 is complete enough for most elementary integrals. Thomas' Calculus §8.1 is largely a practice section: recognising which standard form an integral matches, often after a small algebraic rewrite, before reaching for a heavier technique.
Example 12.4 — Matching a standard form after rewriting
Evaluate ∫ x2+1x dx.
Step 1 — Split the fraction before integrating
x2+1x = x + 1x
Step 2 — Integrate term by term (Eq. 9.5a, Eq. 11.7)
x2/2 + ln|x| + C
12.6 Integration by Parts
Reversing the Product Rule (Week 4 §4.7), d/dx[uv] = u′v + uv′, integrate both sides and rearrange:
Use it when the integrand is a product of two factors, one that simplifies under differentiation (choose as u) and one that is easy to integrate (choose as dv). The mnemonic LIATE (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential) suggests a priority order for choosing u, roughly from "hardest to integrate" to "easiest."
Example 12.5 — Integration by parts
Evaluate ∫ x ex dx.
Step 1 — Choose u and dv (LIATE: algebraic before exponential)
u=x, dv=exdx ⇒ du=dx, v=ex.
Step 2 — Apply Eq. 12.2
xex − ∫ ex dx
Step 3 — Finish the remaining (easier) integral
xex − ex + C
12.7 Repeated Integration by Parts
Some integrals need the technique applied more than once, or need a self-referential trick where the original integral reappears and can be solved for algebraically.
Example 12.6 — The self-referential trick
Evaluate I = ∫ ex cos x dx.
Step 1 — Apply parts once
u=ex, dv=cos x dx: I = exsin x − ∫ ex sin x dx.
Step 2 — Apply parts again to the remaining integral
u=ex, dv=sin x dx: ∫ exsin x dx = −excos x + ∫ excos x dx = −excos x + I.
Step 3 — Substitute back; I reappears, so solve algebraically
I = exsin x − (−excos x + I) = exsin x + excos x − I
2I = ex(sin x+cos x)
I = (1/2)ex(sin x+cos x) + C
12.8 Trigonometric Integrals: Odd Powers
For ∫ sinmx cosnx dx with at least one of m, n odd: peel off one factor of the odd-power function, convert the rest using sin2+cos2=1 (Week 1, Eq. 1.8), and substitute (Week 9, §9.6) using the remaining single factor.
Example 12.7 — Odd power of sine
Evaluate ∫ sin3x dx.
Step 1 — Peel off one sine factor and convert the rest
sin3x = sin2x sin x = (1−cos2x) sin x.
Step 2 — Substitute u=cos x
du=−sin x dx: ∫ (1−u2)(−du) = ∫ (u2−1) du
Step 3 — Integrate and substitute back
u3/3 − u + C = cos3x/3 − cos x + C
12.9 Trigonometric Integrals: Even Powers
When both m, n are even, no single factor can be peeled off; instead use the half-angle identities
to rewrite the integrand at half the original power, trading one hard integral for a more manageable one, possibly requiring the identities again.
Example 12.8 — Even power of sine
Evaluate ∫ sin2x dx.
Step 1 — Apply Eq. 12.3a
∫ 1−cos 2x2 dx
Step 2 — Integrate term by term
x/2 − sin(2x)/4 + C
12.10 Worked Capstone Example
Example 12.9 — Integration by parts with a logarithm
Evaluate ∫1e ln x dx, combining Week 11's logarithm with this week's technique.
Step 1 — Choose u and dv (LIATE: logarithmic first)
u=ln x, dv=dx ⇒ du=(1/x)dx, v=x.
Step 2 — Apply Eq. 12.2
x ln x − ∫ x·(1/x) dx = x ln x − ∫ dx = x ln x − x + C
Step 3 — Evaluate as a definite integral (FTC Part 2, Week 9)
[x ln x − x]1e = (e·1 − e) − (0 − 1)
= 0 − (−1) = 1
12.11 Nomenclature
| Symbol | Meaning |
|---|---|
| indeterminate form | A limit expression like 0/0 or ∞/∞ that gives no information until resolved |
| LIATE | Priority order for choosing u in integration by parts: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential |
| u, dv | The two pieces chosen when applying integration by parts |
12.12 Problems
- L'Hôpital's, 0/0. Evaluate limx→0 (sin x)/x, and compare to the special limit used in Week 4.
- L'Hôpital's, ∞/∞. Evaluate limx→∞ (ln x)/x.
- Repeated L'Hôpital's. Evaluate limx→0 (x−sin x)/x3.
- Indeterminate product. Evaluate limx→∞ x sin(1/x).
- Indeterminate power. Evaluate limx→∞ (1+1/x)x.
- Integration by parts. Evaluate ∫ x cos x dx.
- Integration by parts, twice. Evaluate ∫ x2ex dx.
- Odd power. Evaluate ∫ cos3x dx.
- Even power. Evaluate ∫ cos2x dx.
- Combined. Evaluate ∫ sin2x cos3x dx (one power is odd, one even — use §12.8's method).
12.13 References
- Thomas, G. B., Weir, M. D. and Hass, J. Thomas' Calculus, SI Units, 15th ed. Pearson, 2024. — §7.5 (indeterminate forms and L'Hôpital's rule), §8.1 (using basic integration formulas), §8.2 (integration by parts), §8.3 (trigonometric integrals).
- Pearson MyLab Math — the Week 12 homework set is assigned directly from the sections above; see the Calendar for the due date.