Rucete ✏ AP Calculus In a Nutshell
5. Antidifferentiation
This chapter introduces indefinite integrals, antiderivative rules, and basic integration techniques, including substitution. BC students also cover integration by parts and partial fractions.
Antiderivatives
• The antiderivative of f(x) is a function F(x) such that F′(x) = f(x)
• General form: ∫f(x) dx = F(x) + C (C is the constant of integration)
• If two functions have the same derivative, they differ by a constant
Basic Formulas
Common antiderivative rules (let n ≠ –1):
• ∫xⁿ dx = (1/(n + 1)) xⁿ⁺¹ + C
• ∫1/x dx = ln|x| + C
• ∫eˣ dx = eˣ + C
• ∫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
Substitution Technique
• Let u = inner function, so du = derivative of u
• Rewrite integral in terms of u
Example: ∫2(1 – 3x)² dx Let u = 1 – 3x → du = –3 dx → ∫2u²(–1/3) du = –(2/3) ∫u² du = –(2/9)u³ + C → –(2/9)(1 – 3x)³ + C
Rational Functions
• If numerator degree ≥ denominator degree → use long division
• Example: ∫(2x² + 3x + 1)/(x + 1) dx → divide first
Antiderivatives of Trig and Exponential Combinations
• Many require substitution
• Use identities if needed: – sin(2x) = 2 sin x cos x – 1 + tan²x = sec²x
Example: ∫ex tan ex dx → let u = ex → du = ex dx → ∫tan u du = ln|sec u| + C = ln|sec ex| + C
Antiderivatives with Chain Rule Reversal
• If f(x) = g(h(x)) · h′(x), then – ∫f(x) dx = G(h(x)) + C, where G is antiderivative of g
Example: ∫(2x³)(2x³ – 1)² dx Let u = 2x³ – 1 → du = 6x² dx → Express in terms of u and integrate
Integration by Parts (BC)
• Formula: ∫u dv = uv – ∫v du
• Use when product of functions is hard to integrate directly
• Choose u and dv wisely using LIATE rule: – Logarithmic, Inverse trig, Algebraic, Trig, Exponential
Example: ∫x eˣ dx Let u = x → du = dx dv = eˣ dx → v = eˣ → ∫x eˣ dx = x eˣ – ∫eˣ dx = x eˣ – eˣ + C
Tic-Tac-Toe Method (Integration by Parts Shortcut)
• Use when one factor differentiates to 0 (like a polynomial)
• Create columns for u, dv → alternate signs down the table
• Multiply diagonally and sum
Example: ∫x² eˣ dx → u = x² → 2x → 2 → 0 → dv = eˣ → eˣ → eˣ → eˣ → ∫x² eˣ dx = x² eˣ – 2x eˣ + 2eˣ + C
Partial Fractions (BC)
• Use when integrating rational functions where degree(top) < degree(bottom)
• Step-by-step:
1. Factor denominator
2. Split into sum of simpler fractions
3. Solve for constants using algebra
4. Integrate each term
Example: ∫(1/(x² – 1)) dx = ∫[1/(x – 1) – 1/(x + 1)]/2 dx → (1/2)ln|x – 1| – (1/2)ln|x + 1| + C
Motion Problems with Antiderivatives
• If v(t) = velocity, then s(t) = ∫v(t) dt + C
• If a(t) = acceleration, then v(t) = ∫a(t) dt + C
• Use given initial conditions to find constants
Example: Given a(t) = –32, v(0) = 64 → v(t) = –32t + 64 s(0) = 80 → s(t) = –16t² + 64t + 80
Using Initial Conditions
• Given f′(x) and a value f(a) = b, – Integrate f′(x) to get f(x) + C – Plug in a and b to solve for C
Example: f′(x) = 3x² – 4x, f(1) = 2 → f(x) = x³ – 2x² + C → 1 – 2 + C = 2 → C = 3 → f(x) = x³ – 2x² + 3
In a Nutshell
Antidifferentiation allows us to recover original functions from derivatives. Mastery includes basic rules, substitution, and BC techniques like integration by parts and partial fractions. These tools solve real-world problems like motion and modeling when combined with initial conditions.