Calculus 2 – Complete Reference Guide

1. U-Substitution Review

Method: $\int f(g(x))g'(x)dx$

Let $u = g(x)$, then $du = g'(x)dx$

$\int f(u) du = F(u) + C = F(g(x)) + C$

Example

$\int 2x e^{x^2} dx = \int e^u du = e^u + C = e^{x^2} + C$

Choose $u$ to simplify the integrand

2. Integration by Parts

Formula: $\int u \, dv = uv - \int v \, du$

LIATE Rule

Log, Inverse trig, Algebraic, Trig, Exponential

Tabular Method

Use when integrating polynomial × trig/exp

Alternate signs: + − + − ...

Sum products along diagonals

3. Trigonometric Integrals

Powers of Sin & Cos

Odd power: Factor out one, use $\sin^2 + \cos^2 = 1$

Even power: $\sin^2 x = \frac{1-\cos 2x}{2}$

Even power: $\cos^2 x = \frac{1+\cos 2x}{2}$

Tan & Sec

$\int \tan^m x \sec^n x \, dx$

If $n$ even: factor out $\sec^2 x$, use $\sec^2 - 1 = \tan^2$

If $m$ odd: factor out $\tan x \sec x$

4. Trigonometric Substitution

$\sqrt{a^2-x^2} \Rightarrow x = a\sin\theta$, $dx = a\cos\theta \, d\theta$

$\sqrt{a^2+x^2} \Rightarrow x = a\tan\theta$, $dx = a\sec^2\theta \, d\theta$

$\sqrt{x^2-a^2} \Rightarrow x = a\sec\theta$, $dx = a\sec\theta\tan\theta \, d\theta$

Key Identity

$\sin^2\theta + \cos^2\theta = 1$

$\sec^2\theta - \tan^2\theta = 1$

Draw right triangle to convert back

5. Partial Fractions

Linear Factors

$\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$

Repeated Factors

$\frac{P(x)}{(x-a)^n} = \frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \cdots + \frac{A_n}{(x-a)^n}$

Quadratic Factors

$\frac{P(x)}{(x^2+bx+c)(x-a)} = \frac{Ax+B}{x^2+bx+c} + \frac{C}{x-a}$

6. Improper Integrals

Type 1: Infinite Limits

$\int_a^{\infty} f(x)dx = \lim_{t \to \infty} \int_a^t f(x)dx$

Type 2: Discontinuities

If discontinuous at $x=c$ in $[a,b]$:

$\int_a^b f(x)dx = \int_a^c f(x)dx + \int_c^b f(x)dx$ (both as limits)

Convergent if limit exists and is finite

7. Sequences

Convergence

$\lim_{n \to \infty} a_n = L$ (sequence converges to $L$)

Properties

Monotonic: Always increasing or decreasing

Bounded: $m \leq a_n \leq M$ for all $n$

Monotone Convergence: Monotonic + bounded = convergent

Divergence

If $\lim_{n \to \infty} a_n$ does not exist, the sequence diverges

8. Series Basics

Partial Sums

$S_n = \sum_{i=1}^n a_i$

Series Convergence

$\sum_{n=1}^{\infty} a_n = \lim_{n \to \infty} S_n$

Geometric Series

$\sum ar^n$ converges to $\frac{a}{1-r}$ if $|r| < 1$

Telescoping Series

$\sum (a_n - a_{n+1})$: most terms cancel

9. Integral Test

If $f$ continuous, positive, decreasing:

$\sum_{n=1}^{\infty} a_n$ and $\int_1^{\infty} f(x)dx$ either both converge or both diverge

P-Series

$\sum \frac{1}{n^p}$ converges if $p > 1$

$\sum \frac{1}{n^p}$ diverges if $p \leq 1$

Useful for determining series behavior

10. Comparison Tests

Direct Comparison

If $0 \leq a_n \leq b_n$:

$\sum b_n$ converges $\Rightarrow \sum a_n$ converges

$\sum a_n$ diverges $\Rightarrow \sum b_n$ diverges

Limit Comparison

$L = \lim \frac{a_n}{b_n}$ (L > 0, finite)

Then $\sum a_n$ and $\sum b_n$ both converge or both diverge

11. Alternating Series Test

For $\sum (-1)^n a_n$:

$a_n$ is decreasing

$\lim a_n = 0$

$\Rightarrow$ Series converges

Absolute vs Conditional

Absolutely convergent: $\sum |a_n|$ converges

Conditionally convergent: $\sum a_n$ converges but $\sum |a_n|$ diverges

12. Ratio & Root Tests

Ratio Test

$L = \lim_{n \to \infty} \left|\frac{a_{n+1}}{a_n}\right|$

$L < 1$: Converges absolutely

$L > 1$: Diverges

$L = 1$: Inconclusive

Root Test

$L = \lim_{n \to \infty} \sqrt[n]{|a_n|}$

Same conclusions as Ratio Test

13. Power Series

$\sum_{n=0}^{\infty} c_n(x-a)^n$

Radius of Convergence

Use Ratio Test on $c_n(x-a)^n$

Interval of Convergence

$(a-R, a+R)$

Check endpoints separately

At least center point $x=a$ always works

14. Taylor Series

General Formula

$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$

Maclaurin Series

Taylor series centered at $a = 0$

$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$

15. Common Taylor Series

$e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$

$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \cdots$

$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \cdots$

$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ ($-1 < x \leq 1$)

$\frac{1}{1-x} = 1 + x + x^2 + x^3 + \cdots$ ($|x| < 1$)

16. Arc Length & Surface Area

Arc Length

$L = \int_a^b \sqrt{1 + [f'(x)]^2} \, dx$

Surface Area of Revolution

Rotated around x-axis:

$S = 2\pi \int_a^b f(x)\sqrt{1 + [f'(x)]^2} \, dx$

Rotated around y-axis:

$S = 2\pi \int_a^b x\sqrt{1 + [f'(x)]^2} \, dx$

17. Parametric Equations

Parametric Form

$x = x(t)$, $y = y(t)$

Derivatives

$\frac{dy}{dx} = \frac{dy/dt}{dx/dt}$

Arc Length

$L = \int_a^b \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$

Surface Area

$S = 2\pi \int_a^b y(t)\sqrt{(x'(t))^2 + (y'(t))^2} \, dt$

18. Polar Coordinates & Curves

Polar Form

$r = f(\theta)$

Conversion

$x = r\cos\theta$, $y = r\sin\theta$

$r^2 = x^2 + y^2$, $\tan\theta = \frac{y}{x}$

Area

$A = \frac{1}{2}\int_\alpha^\beta r^2 \, d\theta$

Arc Length

$L = \int_\alpha^\beta \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta$

CALCULUS 2 Formula Sheet