AP Calculus BC Unit 6: Integration Techniques (Concepts, Methods, and Common Pitfalls)

Antiderivative

A function F(x) whose derivative is f(x); i.e., F'(x)=f(x).

Indefinite Integral

Notation for the family of all antiderivatives: ∫ f(x) dx = F(x) + C.

Constant of Integration (C)

An arbitrary constant added to an indefinite integral because derivatives of constants are 0, so all antiderivatives differ by a constant.

Definite Integral via Antiderivative

Net accumulation from a to b computed using an antiderivative: ∫_a^b f(x) dx = F(b) − F(a).

Linearity of Integration

You can integrate term-by-term and pull out constants: ∫(af(x)+bg(x)) dx = a∫f(x) dx + b∫g(x) dx.

Power Rule for Antiderivatives (n ≠ −1)

For n≠−1, ∫ x^n dx = x^(n+1)/(n+1) + C.

Logarithm Exception (n = −1)

Because d/dx(ln|x|)=1/x, we have ∫(1/x) dx = ln|x| + C (not a power-rule result).

Absolute Value in ln|x|

The bars are required because ln|x| differentiates to 1/x for both positive and negative x (x≠0).

Antiderivative of e^x

∫ e^x dx = e^x + C.

Antiderivative of a^x

For a>0, a≠1: ∫ a^x dx = a^x/ln(a) + C.

Antiderivative of cos x

∫ cos x dx = sin x + C.

Antiderivative of sin x

∫ sin x dx = −cos x + C.

Secant-Squared Recognition

∫ sec^2 x dx = tan x + C.

Arctangent Recognition Form

∫ 1/(1+x^2) dx = arctan x + C.

Arcsine Recognition Form

∫ 1/√(1−x^2) dx = arcsin x + C.

Substitution (u-Substitution)

A reverse chain-rule method using u=g(x), du=g'(x)dx to turn ∫ f(g(x))g'(x) dx into ∫ f(u) du.

Change of Limits in Substitution

For definite integrals, you may convert x-limits to u-limits using u=g(x) and evaluate entirely in u.

Improper Rational Function

A rational function P(x)/Q(x) where deg(numerator) ≥ deg(denominator); it must be rewritten (typically by long division) before integrating.

Polynomial Long Division (for integrals)

Rewrites P(x)/Q(x) as S(x) + R(x)/Q(x), where S is a polynomial and deg(R)<deg(Q), making the integral manageable.

Completing the Square

Rewriting x^2+bx+c as (x+b/2)^2 + (c−b^2/4) to match standard integral forms (often involving arctan).

Shifted Arctan Form

∫ 1/((x−h)^2 + a^2) dx = (1/a) arctan((x−h)/a) + C.

Integration by Parts

A reverse product-rule technique: ∫ u dv = uv − ∫ v du.

LIATE Heuristic

A guideline for choosing u in integration by parts: Logarithmic, Inverse trig, Algebraic, Trig, Exponential (choose u earlier when possible).

Partial Fraction Decomposition (Linear Factors)

Rewriting a proper rational function into a sum like A/(x−a)+B/(x−b) so each term integrates to logs/powers.

Improper Integral

A definite integral with an infinite interval and/or an integrand that becomes infinite (vertical asymptote); evaluated using limits to test convergence/divergence.