Math 31 – Calc II Exam 2 Flashcards

Flashcards based on the Math 31 Calculus II Unit 2 lecture transcript covering integration techniques, convergence tests, and series definitions.

LIATE/LIPET

The priority order for picking $u$ in integration by parts: Logarithmic, Inverse trig, Algebraic/polynomial, Trigonometric, and Exponential.

Strategy for $\int \tan^m(x) \sec^n(x) dx$ ($n$ even)

Save one $\sec^2(x)$ for $du$. Let $u = \tan(x)$ ($du = \sec^2(x) dx$) and convert remaining $\sec^2(x)$ factors using $\sec^2(x) = 1 + \tan^2(x)$.

Strategy for $\int \tan^m(x) \sec^n(x) dx$ ($m$ odd)

Save one $\sec(x)\tan(x)$ for $du$. Let $u = \sec(x)$ ($du = \sec(x)\tan(x) dx$) and convert remaining $\tan^2(x)$ using $\tan^2(x) = \sec^2(x) - 1$.

Trig Substitution for $\sqrt{a^2 - x^2}$

Use $x = a \sin(\theta)$, $dx = a \cos(\theta) d\theta$, which simplifies to $a \cos(\theta)$.

Trig Substitution for $\sqrt{a^2 + x^2}$

Use $x = a \tan(\theta)$, $dx = a \sec^2(\theta) d\theta$, which simplifies to $a \sec(\theta)$.

Trig Substitution for $\sqrt{x^2 - a^2}$

Use $x = a \sec(\theta)$, $dx = a \sec(\theta) \tan(\theta) d\theta$, which simplifies to $a \tan(\theta)$.

Completing the Square in Integration

A technique used when a quadratic under a radical has a linear term (e.g., $8 - 2x - x^2$) before applying a trig substitution.

Polynomial Long Division Requirement

Must be performed before partial fraction decomposition when the degree of the numerator is greater than or equal to the degree of the denominator ($\text{degree(numerator)} \geq \text{degree(denominator)}$).

Decomposition for Distinct Linear Factors

The form $\frac{A}{x} + \frac{B}{x-3} + \frac{C}{x+3}$ for a denominator like $x(x-3)(x+3)$.

Decomposition for Repeated Linear Factor $(x-a)^2$

The form $\frac{A}{x-a} + \frac{B}{(x-a)^2}$.

Decomposition for Irreducible Quadratic Factor

For a denominator like $(x-a)(x^2+b^2)$, the form is $\frac{A}{x-a} + \frac{Bx+C}{x^2+b^2}$.

Numerical Integration Rules

Trapezoidal Rule ($T_n$), Simpson’s Rule ($S_n$), and Midpoint Rule ($M_n$) used to approximate integrals over subintervals.

Simpson's Rule Special Requirement

The number of subintervals $n$ must be even.

Improper Integral Types

Type 1 integrals have infinite limits of integration; Type 2 integrals have a discontinuous integrand on the interval.

Improper Integral Convergence

Occurs when the relevant limit exists and equals a finite real number $L$. If the limit does not exist or goes to $\pm\infty$, it diverges.

Sequence Convergence

A sequence $\left\{ a_n \right\}$ converges if $\lim_{n \to \infty} a_n = L$ for some finite real number $L$.

Order of Growth Rates (Slowest to Fastest)

$\ln(n) < n^p \, (p>0) < n < n^x \, (x>1) < x^n < n! < n^n$. This hierarchy is used for evaluating limits as $n \to \infty$.

Infinite Series Sum Definition

$\sum a_n = \lim_{k \to \infty} S_k$, where $S_k = \sum_{n=1}^{k} a_n$ is the $k$-th partial sum.

Absolute Convergence

A series $\sum a_n$ is absolutely convergent if the series of absolute values $\sum |a_n|$ converges.

Conditional Convergence

A series $\sum a_n$ converges, but $\sum |a_n|$ diverges.

Absolute Convergence Theorem

If a series is absolutely convergent, then it is convergent ($\text{Absolute convergence} \Longrightarrow \text{convergence}$).

n-th Term Test for Divergence

If $\lim_{n \to \infty} a_n \neq 0$, the series diverges. If the limit is $0$, the test is inconclusive.

Geometric Series Test

A series $\sum ar^n$ converges to $S = \frac{a}{1-r}$ if $|r| < 1$ and diverges if $|r| \geq 1$.

p-Series Test

A series $\sum \frac{1}{n^p}$ converges if $p > 1$ and diverges if $p \leq 1$ (e.g., the harmonic series $\sum \frac{1}{n}$ diverges).

Integral Test

If $a_n = f(n)$ with $f$ being positive, continuous, and decreasing for $x \geq 1$, then $\sum a_n$ and $\int_{1}^{\infty} f(x) dx$ converge or diverge together.

Direct Comparison Test

For $0 \leq a_n \leq b_n$: if $\sum b_n$ converges, $\sum a_n$ converges; if $\sum a_n$ diverges, $\sum b_n$ diverges.

Limit Comparison Test

Given $L = \lim_{n \to \infty} \frac{a_n}{b_n}$ where $a_n, b_n > 0$, if $0 < L < \infty$, both series behave the same.

Alternating Series Test Conditions

For $\sum (-1)^n b_n$ with $b_n > 0$, the series converges if (1) $b_{n+1} \leq b_n$ (decreasing) and (2) $\lim_{n \to \infty} b_n = 0$.

Ratio Test

A test where $L = \lim_{n \to \infty} |\frac{a_{n+1}}{a_n}|$. $L < 1$ implies absolute convergence, $L > 1$ divergence, and $L = 1$ is inconclusive. Best for factorials and exponentials.

Root Test

A test where $L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$. Best used when the entire term is raised to the $n$-th power; conclusions match the Ratio Test.