Calculus 1AA3 – Integration Techniques & Improper Integrals

Vocabulary flashcards covering major concepts from the lecture notes on integration techniques and improper integrals.

Improper Integral

An integral whose interval is unbounded or whose integrand becomes infinite within the interval.

Type I Improper Integral

An integral where the interval extends to ±∞ (e.g., ∫ₐ^∞ f(x) dx).

Type II Improper Integral

An integral with a finite interval but an integrand that blows up at an endpoint (e.g., ∫ₐ^ᵇ f(x) dx where f is unbounded at a or b).

Convergent Integral

An improper integral whose limit exists and equals a finite real number.

Divergent Integral

An improper integral whose limit does not exist or is infinite.

p-Integral (Type I)

∫₁^∞ x^(–p) dx, convergent for p > 1 and divergent for p ≤ 1.

p-Integral (Type II)

∫₀¹ x^(–p) dx, convergent for p < 1 and divergent for p ≥ 1.

Comparison Test (Improper Integrals)

Uses inequalities between functions to infer convergence or divergence: ‘greater than divergent diverges; less than convergent converges.’

Integration by Parts

Technique based on ∫u dv = u v – ∫v du; choose u from ‘LIATE’ preferences (logs, inverse trig, algebraic, trig, exponential).

u-Substitution

Change of variables method where u = g(x) simplifies the integrand and du replaces g'(x) dx.

Trig Substitution

Replaces x with a trig function (x = a sin t, a tan t, or a sec t) to simplify integrals containing √(a²–x²), √(a²+x²), or √(x²–a²).

Partial Fractions Decomposition

Expresses a rational function P(x)/Q(x) as a sum of simpler fractions to integrate term-by-term.

Hyperbolic Sine (sinh x)

Defined as (eˣ – e^(–x))/2; satisfies d/dx sinh x = cosh x and ∫sinh x dx = cosh x + C.

Hyperbolic Cosine (cosh x)

Defined as (eˣ + e^(–x))/2; satisfies d/dx cosh x = sinh x and ∫cosh x dx = sinh x + C.

Hyperbolic Tangent (tanh x)

Given by sinh x / cosh x; derivative is sech² x.

Trig Identity for sin²

sin² t = (1 – cos 2t)/2, useful for reducing even powers in integrals.

Trig Identity for cos²

cos² t = (1 + cos 2t)/2, used to integrate even cosine powers.

Reduction Formula for sec x

∫sec x dx = ln|sec x + tan x| + C, often appears when no simpler substitution works.

Even-Odd Strategy (sinᵐ x cosⁿ x)

When one power is odd, split off one factor to form du; when both are even, use half-angle identities.

Even-Odd Strategy (tanᵐ x secⁿ x)

If n is even, set u = tan x; if m is odd, set u = sec x; otherwise convert with sec² x = 1 + tan² x.

Extended Type I Integral

For ∫{–∞}^{∞} f(x) dx, defined as the sum of limits ∫{–∞}^a + ∫_a^{∞}; both must converge.

Limit Definition of Improper Integral

Replaces the problematic bound with a variable (b or t), integrates on a finite interval, and takes the limit.

LIATE Rule

Guideline for choosing u in integration by parts: Logarithmic, Inverse trig, Algebraic, Trig, Exponential.

Cauchy Principal Value (mention)

A symmetric limit used when separate one-sided limits diverge; not accepted for convergence in this course.

Course Grading (1AA3)

Assignments 20%, Test 1 20%, Test 2 20%, Final Exam 40% – total 100%.

Improper Integral ‘Tail’ Principle

For Type I, convergence depends only on behaviour as x → ±∞; finite intervals don’t affect convergence.

Integration by Trigonometric Identities

Simplifies integrals by converting products/powers of trig functions using identities before integrating.