Calculus 2 Complete Course Cheat Sheet

Module 1: Techniques of Integration

Substitution Rule

∫ f(g(x))g'(x) dx = ∫ f(u) du, where u = g(x)

U-Substitution Steps:

Choose u = g(x) (usually the "inside" function)
Find du = g'(x) dx
Substitute u and du into integral
Integrate with respect to u
Substitute back: replace u with g(x)

Example:

∫ 2x cos(x²) dx
Let u = x², then du = 2x dx
∫ cos(u) du = sin(u) + C = sin(x²) + C

Integration by Parts

∫ u dv = uv - ∫ v du
where u and dv are chosen parts of the integrand

LIATE Rule for choosing u:
L - Logarithmic (ln x, log x)
I - Inverse trig (arcsin x, arctan x)
A - Algebraic (x, x², polynomials)
T - Trigonometric (sin x, cos x)
E - Exponential (e^x, a^x)

Integration by Parts Steps:

Choose u (using LIATE) and dv
Find du and v
Apply formula: uv - ∫ v du
Integrate ∫ v du

Example:

∫ x e^x dx
u = x, dv = e^x dx
du = dx, v = e^x
= x e^x - ∫ e^x dx = x e^x - e^x + C = e^x(x-1) + C

Trigonometric Integrals

Powers of sin and cos

∫ sin^m x cos^n x dx

• If m odd: save one sin, convert rest using sin²x = 1-cos²x
• If n odd: save one cos, convert rest using cos²x = 1-sin²x
• If both even: use half-angle formulas

Half-Angle Formulas

sin²x = (1 - cos 2x)/2
cos²x = (1 + cos 2x)/2
sin x cos x = (sin 2x)/2

Powers of tan and sec

∫ tan^m x sec^n x dx

• If n even: save sec²x, convert rest using sec²x = 1+tan²x
• If m odd: save sec x tan x, convert rest

Trigonometric Substitution

√(a² - x²)

x = a sin θ
dx = a cos θ dθ
√(a² - x²) = a cos θ

√(a² + x²)

x = a tan θ
dx = a sec² θ dθ
√(a² + x²) = a sec θ

√(x² - a²)

x = a sec θ
dx = a sec θ tan θ dθ
√(x² - a²) = a tan θ

Partial Fractions

For rational functions P(x)/Q(x) where degree(P) < degree(Q)

Linear Factors

1/((x-a)(x-b)) = A/(x-a) + B/(x-b)

Repeated Linear

1/(x-a)² = A/(x-a) + B/(x-a)²

Quadratic Factors

1/((x²+1)(x-a)) = (Ax+B)/(x²+1) + C/(x-a)

Partial Fraction Steps:

Factor denominator completely
Write partial fraction decomposition
Multiply both sides by denominator
Solve for coefficients (substitution or comparing coefficients)
Integrate each term

Numerical Integration

Trapezoidal Rule

∫[a to b] f(x) dx ≈ (Δx/2)[f(x₀) + 2f(x₁) + ... + 2f(x_{n-1}) + f(x_n)]
where Δx = (b-a)/n

Simpson's Rule

∫[a to b] f(x) dx ≈ (Δx/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(x_n)]
where n is even

Module 2: Applications of Integration & Sequences

Improper Integrals

Type 1: Infinite Limits

∫[a to ∞] f(x) dx = lim[t→∞] ∫[a to t] f(x) dx
∫[-∞ to b] f(x) dx = lim[t→-∞] ∫[t to b] f(x) dx

Type 2: Discontinuous Integrand

If f has discontinuity at x = c in [a,b]:
∫[a to b] f(x) dx = ∫[a to c] f(x) dx + ∫[c to b] f(x) dx

Comparison Test: If 0 ≤ f(x) ≤ g(x) for x ≥ a:
• If ∫g(x)dx converges → ∫f(x)dx converges
• If ∫f(x)dx diverges → ∫g(x)dx diverges

Areas Between Curves

Area = ∫[a to b] |f(x) - g(x)| dx
where f(x) ≥ g(x) on [a,b]

Steps for Area Between Curves:

Find intersection points (solve f(x) = g(x))
Determine which function is on top
Set up integral(s) with proper limits
Integrate

Volumes

Disk Method

V = π ∫[a to b] [R(x)]² dx
Rotating around x-axis
R(x) = radius function

Washer Method

V = π ∫[a to b] [R(x)]² - [r(x)]² dx
R(x) = outer radius
r(x) = inner radius

Shell Method

V = 2π ∫[a to b] x · f(x) dx
Rotating around y-axis
x = radius of shell
f(x) = height of shell

When to use Shell vs Disk/Washer:
Use shell method when setup is simpler (fewer integrals needed)

Arc Length

Arc Length = ∫[a to b] √(1 + [f'(x)]²) dx

For parametric: L = ∫[a to b] √((dx/dt)² + (dy/dt)²) dt

Physics & Engineering Applications

Work

W = ∫[a to b] F(x) dx
where F(x) is force function

Fluid Pressure

F = ∫[a to b] ρg · depth(y) · width(y) dy
ρ = fluid density
g = gravitational constant

Center of Mass

x̄ = (1/m) ∫[a to b] x · ρ(x) dx
ȳ = (1/m) ∫[a to b] y · ρ(y) dy
where m = total mass

Sequences

Sequence {aₙ} converges to L if lim[n→∞] aₙ = L

Common Limits

lim[n→∞] 1/nᵖ = 0 (p > 0)
lim[n→∞] rⁿ = 0 (|r| < 1)
lim[n→∞] rⁿ = ∞ (r > 1)
lim[n→∞] (1 + 1/n)ⁿ = e

Squeeze Theorem

If aₙ ≤ bₙ ≤ cₙ and
lim aₙ = lim cₙ = L
then lim bₙ = L

Module 3: Series

Series Basics

Infinite Series: ∑[n=1 to ∞] aₙ = a₁ + a₂ + a₃ + ...
Converges if lim[n→∞] Sₙ exists (finite)
where Sₙ = ∑[k=1 to n] aₖ (partial sum)

Geometric Series:
∑[n=0 to ∞] arⁿ = a/(1-r) if |r| < 1
Diverges if |r| ≥ 1

Convergence Tests

nth Term Test

If lim[n→∞] aₙ ≠ 0

Then ∑aₙ diverges
(Note: If lim aₙ = 0, test inconclusive)

Integral Test

f(x) positive, decreasing, continuous

∑f(n) and ∫f(x)dx both converge or both diverge

Comparison Test

0 ≤ aₙ ≤ bₙ for all n

If ∑bₙ converges → ∑aₙ converges
If ∑aₙ diverges → ∑bₙ diverges

Limit Comparison

L = lim[n→∞] aₙ/bₙ

If 0 < L < ∞, then ∑aₙ and ∑bₙ have same behavior

Ratio Test

L = lim[n→∞] |aₙ₊₁/aₙ|

L < 1: converges absolutely
L > 1: diverges
L = 1: inconclusive

Root Test

L = lim[n→∞] ⁿ√|aₙ|

L < 1: converges absolutely
L > 1: diverges
L = 1: inconclusive

Alternating Series

∑(-1)ⁿbₙ where bₙ > 0, decreasing, lim bₙ = 0

Converges
Error: |Rₙ| ≤ bₙ₊₁

Power Series

∑[n=0 to ∞] cₙ(x-a)ⁿ = c₀ + c₁(x-a) + c₂(x-a)² + ...

Radius of Convergence:
R = lim[n→∞] |cₙ/cₙ₊₁| or R = 1/lim[n→∞] |cₙ₊₁/cₙ|
Interval: (a-R, a+R) - check endpoints separately!

Operations on Power Series

Addition/Subtraction: term by term
Differentiation: ∑ncₙxⁿ⁻¹
Integration: ∑cₙxⁿ⁺¹/(n+1)

Taylor & Maclaurin Series

Taylor Series (about x = a):
f(x) = ∑[n=0 to ∞] f⁽ⁿ⁾(a)/n! · (x-a)ⁿ

Maclaurin Series (a = 0):
f(x) = ∑[n=0 to ∞] f⁽ⁿ⁾(0)/n! · xⁿ

Common Maclaurin Series

eˣ = ∑xⁿ/n! = 1 + x + x²/2! + x³/3! + ...

sin x = ∑(-1)ⁿx²ⁿ⁺¹/(2n+1)! = x - x³/3! + x⁵/5! - ...

cos x = ∑(-1)ⁿx²ⁿ/(2n)! = 1 - x²/2! + x⁴/4! - ...

More Series

1/(1-x) = ∑xⁿ = 1 + x + x² + ... (|x| < 1)

ln(1+x) = ∑(-1)ⁿ⁺¹xⁿ/n = x - x²/2 + x³/3 - ...

(1+x)ᵏ = ∑(k choose n)xⁿ

Module 4: Parametric & Polar Curves

Parametric Curves

Parametric Equations: x = f(t), y = g(t)
Curve traced as parameter t varies

Eliminating Parameter

Solve one equation for t
Substitute into other equation
Get relation y = F(x)

Derivatives

dy/dx = (dy/dt)/(dx/dt)

d²y/dx² = d/dx[dy/dx] = (d/dt[dy/dx])/(dx/dt)

Calculus with Parametric Curves

Arc Length

L = ∫[α to β] √((dx/dt)² + (dy/dt)²) dt

Surface Area

S = 2π ∫[α to β] y √((dx/dt)² + (dy/dt)²) dt
(rotation around x-axis)

Polar Coordinates

Polar: (r, θ) where r ≥ 0, 0 ≤ θ < 2π
Conversion: x = r cos θ, y = r sin θ
r² = x² + y², θ = arctan(y/x)

Common Polar Curves

Circle: r = a
Line through origin: θ = α
Cardioid: r = a(1 ± cos θ)
Rose: r = a cos(nθ) or r = a sin(nθ)

Tangent Lines

dy/dx = (dr/dθ sin θ + r cos θ)/(dr/dθ cos θ - r sin θ)

Areas & Lengths in Polar

Area in Polar

A = (1/2) ∫[α to β] r² dθ

Between curves:
A = (1/2) ∫[α to β] [r₁² - r₂²] dθ

Arc Length in Polar

L = ∫[α to β] √(r² + (dr/dθ)²) dθ

Quick Reference & Common Integrals

Basic Integrals

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C
∫ 1/x dx = ln|x| + C
∫ eˣ dx = eˣ + C
∫ aˣ dx = aˣ/ln a + C

Trigonometric

∫ sin x dx = -cos x + C
∫ cos x dx = sin x + C
∫ sec²x dx = tan x + C
∫ sec x tan x dx = sec x + C

Inverse Trig

∫ 1/√(a²-x²) dx = arcsin(x/a) + C
∫ 1/(a²+x²) dx = (1/a)arctan(x/a) + C
∫ 1/(x√(x²-a²)) dx = (1/a)arcsec(|x|/a) + C

Strategy Flowchart

1. Try basic antiderivatives
2. Try u-substitution
3. Try integration by parts
4. Try trig substitution
5. Try partial fractions
6. Look up in tables

Test Strategy:
• Always check if direct integration works first
• For products: try u-sub first, then integration by parts
• For rational functions: partial fractions
• For radicals: trigonometric substitution
• For series: determine convergence before finding sum