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≠−1: ∫x^n dx = x^(n+1)/(n+1) + C.

Special case n = −1 (log rule)

∫(1/x) dx = ln|x| + C (absolute value gives a correct derivative for x

Linearity (sum rule)

∫(f(x)+g(x))dx = ∫f(x)dx + ∫g(x)dx.

Constant multiple rule

∫k f(x)dx = k∫f(x)dx for constant k.

Integral of e^x

∫e^x dx = e^x + C.

Integral of a^x

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

Integral of cos(x)

∫cos(x) dx = sin(x) + C.

Integral of sin(x)

∫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 P(x)/Q(x) as S(x) + R(x)/Q(x), where S is the quotient polynomial and deg(R)<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: ∫ 1/(x^2+a^2) dx = (1/a) arctan(x/a) + C; shifted: ∫1/((x−h)^2+a^2)dx=(1/a)arctan((x−h)/a)+C.