Unit 6 Integration Tools: Building Antiderivatives and Choosing Techniques

Antiderivative

A function F(x) such that $F'(x) = f(x).$

Indefinite integral

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

Constant of integration (C)

The arbitrary constant added to an antiderivative because (F(x)+C)'=F'(x).

Differential (dx)

Indicates the variable of integration (“with respect to x”); becomes crucial when rewriting integrals in substitution.

Power rule for integrals

For n
eq -1: $\int x^n \, dx = \frac{x^{(n+1)}}{(n+1)} + C.$

Special case n = −1 (log rule)

$\int \frac{1}{x} \, dx = \text{ln}|x| + C$ (absolute value gives a correct derivative for x < 0 and x > 0, x
eq 0).

Linearity (sum rule)

$\int(f(x)+g(x))dx = \int f(x)dx + \int g(x)dx$.

Constant multiple rule

∫k f(x)dx = k∫f(x)dx for constant k.

Integral of e^x

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

Integral of a^x

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

Integral of cos(x)

$\int \cos(x) \,dx = \sin(x) + C$.

Integral of sin(x)

$\int \sin(x) \,dx = -\cos(x) + C$.

Initial condition

Extra information like F(a)=b used to determine the constant C and get a unique antiderivative.

Most general antiderivative

An antiderivative written with +C to represent the entire family of solutions.

u-substitution (substitution)

Technique that reverses the chain rule by letting u=g(x) to simplify ∫f(g(x))g'(x)dx into ∫f(u)du.

Reverse the chain rule pattern

Recognize integrands of the form f(g(x))·g'(x), suggesting u=g(x).

Inner function

The “inside” expression g(x) chosen as u in substitution (often inside parentheses, an exponent, radical, or denominator).

du conversion

If u=g(x), then du=g'(x)dx; you rewrite the entire integral using u and du (no leftover x’s).

Constant factor adjustment in substitution

When du differs by a constant (e.g., du=3dx), you compensate by multiplying by the reciprocal (dx=(1/3)du).

Rational function

A ratio of polynomials, P(x)/Q(x).

Top-heavy rational function

A rational function with deg(P) ≥ deg(Q); you typically perform long division before integrating.

Long division decomposition (quotient + remainder)

Rewrite $\frac{P(x)}{Q(x)}$ as $S(x) + \frac{R(x)}{Q(x)}$, where S is the quotient polynomial and $\text{deg}(R) < \text{deg}(Q).$

Log substitution pattern (f'/f)

If the integrand is f'(x)/f(x), then ∫f'(x)/f(x) dx = ln|f(x)| + C.

Completing the square

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

Arctan integral form

For a > 0: $\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \arctan \left( \frac{x}{a} \right) + C;$ shifted: $\int \frac{1}{(x-h)^2 + a^2} \, dx = \frac{1}{a} \arctan \left( \frac{x-h}{a} \right) + C.$