Calculus II Premium Problem Set

01 Integration by Parts ▾

Derived from the product rule for differentiation. Used when the integrand is a product of two functions.

∫ u dv = uv − ∫ v du

LIATE Rule (choose u in this order):
Logarithm → Inverse trig → Algebraic → Trigonometric → Exponential

Example

∫ x eˣ dx → u = x, dv = eˣdx → du = dx, v = eˣ
= xeˣ − eˣ + C = eˣ(x−1) + C

02 Trigonometric Substitution ▾

Eliminates square roots of quadratic expressions using Pythagorean identities.

√(a²−x²) → x = a sinθ
√(a²+x²) → x = a tanθ
√(x²−a²) → x = a secθ

Example

∫ dx/√(4−x²) → x = 2 sinθ, dx = 2 cosθ dθ
= ∫ 2cosθ dθ / (2 cosθ) = θ + C = arcsin(x/2) + C

03 Partial Fractions ▾

Decomposes rational functions into simpler fractions. Applies when degree of numerator < degree of denominator.

P(x)/[(x−a)(x−b)] = A/(x−a) + B/(x−b)

Key steps: Factor denominator → Write decomposition → Multiply both sides by denominator → Compare coefficients or substitute roots

Example

∫ dx/(x²−1) = ∫ [½/(x−1) − ½/(x+1)] dx
= ½ ln|x−1| − ½ ln|x+1| + C = ½ ln|(x−1)/(x+1)| + C

04 Improper Integrals ▾

Integrals with infinite limits or discontinuous integrands. Evaluated as limits.

∫₁^∞ f(x) dx = lim[t→∞] ∫₁ᵗ f(x) dx

p-integral test: ∫₁^∞ 1/xᵖ dx converges if p > 1, diverges if p ≤ 1
Near 0: ∫₀¹ 1/xᵖ dx converges if p < 1, diverges if p ≥ 1

Example

∫₁^∞ 1/x² dx = lim[t→∞] [−1/x]₁ᵗ = 0 − (−1) = 1 ✓ Converges

05 Sequences & Series Convergence ▾

Ratio Test: L = lim|aₙ₊₁/aₙ|. L<1 → converge; L>1 → diverge; L=1 → inconclusive
Root Test: L = lim ⁿ√|aₙ|. Same conclusion.
Integral Test: ∑aₙ converges ↔ ∫f converges (f positive, decreasing)
Alternating Series Test: ∑(−1)ⁿbₙ converges if bₙ → 0 and bₙ decreasing
Comparison Test: 0 ≤ aₙ ≤ bₙ; if ∑bₙ converges, so does ∑aₙ

Geometric Series (key!)

∑aᵣⁿ converges to a/(1−r) when |r| < 1

06 Power Series & Radius of Convergence ▾

∑cₙ(x−a)ⁿ, Radius R = 1/lim|cₙ₊₁/cₙ|

Find R: Apply ratio test → solve |x−a| < R
Check endpoints: Separately test x = a±R

Key Series to Memorize

1/(1−x) = ∑xⁿ, |x|<1
eˣ = ∑xⁿ/n!, all x
sin x = ∑(−1)ⁿ x²ⁿ⁺¹/(2n+1)!, all x
cos x = ∑(−1)ⁿ x²ⁿ/(2n)!, all x
ln(1+x) = ∑(−1)ⁿ xⁿ⁺¹/(n+1), −1<x≤1

07 Parametric Equations & Arc Length ▾

dy/dx = (dy/dt)/(dx/dt) [provided dx/dt ≠ 0]

Arc Length L = ∫ √[(dx/dt)² + (dy/dt)²] dt

Area under parametric curve:
A = ∫ y (dx/dt) dt

08 Polar Coordinates ▾

Area A = ½ ∫ r² dθ

Arc Length L = ∫ √[r² + (dr/dθ)²] dθ

Convert: x = r cosθ, y = r sinθ, r² = x²+y²
Circles: r = a (centered at origin), r = 2a cosθ (passes through origin)

Calculus II

Core Concepts & Key Formulas

Calculus II — Problem Set Results