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Calc 2 - Complete Study Guide

Calc 2 - Complete Study Guide




CALC 2 - COMPLETE SURVIVAL GUIDE



Integration by Parts

∫u dv = uv - ∫v du


1. LIATE Priority: Logs → Inverse trig → Algebraic → Trig → Exponential



2. When to Use: Product of two different function types (xe^x, x sin x, x ln x)



3. Sometimes Apply Twice: For x²e^x or x² sin x, use parts twice in a row






Trig Integrals and Substitution


Essential Trig Identities:
sin²x + cos²x = 1

tan²x + 1 = sec²x

1 + cot²x = csc²x

sin²x = (1 - cos 2x)/2

cos²x = (1 + cos 2x)/2




1. Odd Powers of Sine/Cosine: Save one factor, convert rest using sin²x + cos²x = 1



2. Even Powers: Use half-angle formulas to reduce power



3. Trig Substitution Patterns:
√(a² - x²) → x = a sin θ

√(a² + x²) → x = a tan θ

√(x² - a²) → x = a sec θ







Partial Fractions


1. Factor Denominator Completely: Find all linear and irreducible quadratic factors



2. Set Up Fractions:
Linear factor (x - a): A/(x - a)

Repeated linear (x - a)²: A/(x - a) + B/(x - a)²

Irreducible quadratic (x² + bx + c): (Ax + B)/(x² + bx + c)




3. Finding Coefficients:
Method 1: Multiply through and equate coefficients

Method 2: Substitute convenient x-values to solve for constants

Method 3: Cover-up method for simple linear factors







Approximate Integrals

Trapezoidal Rule: T_n = (Δx/2)[f(x₀) + 2f(x₁) + 2f(x₂) + ... + 2f(x_{n-1}) + f(x_n)]

Simpson's Rule: S_n = (Δx/3)[f(x₀) + 4f(x₁) + 2f(x₂) + 4f(x₃) + ... + f(x_n)]


1. Δx = (b - a)/n where n is number of subintervals



2. Simpson's Rule: Requires even number of subintervals, more accurate



3. Error Bounds: Simpson's error goes as (Δx)⁴, Trapezoidal as (Δx)²






Improper Integrals

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


1. Type 1: Infinite limits - take limit as you approach infinity



2. Type 2: Discontinuous integrand - take limit as you approach discontinuity



3. Convergence vs Divergence:
Converges: Limit exists and is finite

Diverges: Limit is infinite or doesn't exist

Compare with p-integrals: ∫[1 to ∞] 1/x^p converges if p > 1







Areas Between Curves

A = ∫[a to b] |f(x) - g(x)| dx


1. Always Sketch: Determine which function is on top in each interval



2. Find Intersection Points: Solve f(x) = g(x) to find limits of integration



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