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: $\int f(x) dx = F(x) + C.$

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: $\int_a^b f(x) dx = F(b) − F(a).$

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 $n≠−1$, $\int x^n dx = \frac{x^{(n+1)}}{(n+1)} + C.$

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

$$\int e^x dx = e^x + C.$$

Antiderivative of a^x

For $a>0$, $a≠1$: $\int a^x dx = \frac{a^x}{\ln(a)} + C.$

Antiderivative of cos x

∫ cos x dx = sin x + C.

Antiderivative of sin x

$\int \, \text{sin} \, x \, dx = -\text{cos} \, x + C$.

Secant-Squared Recognition

$$\int \sec^2 x dx = \tan x + C.$$

Arctangent Recognition Form

$$\int \frac{1}{1+x^2} dx = \arctan x + C.$$

Arcsine Recognition Form

$$\int \frac{1}{\sqrt{1−x^2}} dx = \arcsin x + C.$$

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

$$\int \frac{1}{((x−h)^2 + a^2)} dx = \frac{1}{a} \arctan\left(\frac{x−h}{a}\right) + C.$$

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.