6.14 Selecting Techniques for Antidifferentiation

By this point the unit has built a full toolkit: basic rules, substitution, long division, completing the square and, for BC, integration by parts and partial fractions. On an exam nobody tells you which one to use. This topic is about reading the structure of an integrand and choosing the technique that fits it.

Selecting an antidifferentiation technique

Topic 6.14 Selecting Techniques for Antidifferentiation belongs to FUN-6 and emphasizes Skill 1.C: Identify an appropriate mathematical rule or procedure based on the classification of a given expression. The central question is not “Which formula have I just practiced?” but “What structural feature does this integrand have?”

Use the following classification pipeline:

Form of the integrand	First procedure to consider
Sum of powers, constants, or basic functions	Basic antiderivative rules
A composite expression multiplied by its inner derivative	Substitution
A product such as polynomial times exponential or logarithmic function	Integration by parts
A rational function with a denominator that factors into linear terms	Partial fractions, BC only
A rational function with numerator degree at least the denominator degree	Long division first
A quadratic denominator	Completing the square or a related standard form
A definite integral with a singularity or infinite endpoint	Improper-integral limit

For example,

$$ \int 2x\cos(x^2),dx $$

is classified as a composite-function pattern: the inner function is $u=x^2$, and $du=2x,dx$. Thus,

$$ \int 2x\cos(x^2),dx

\sin(x^2)+C. $$

By contrast,

$$ \int x e^x,dx $$

is a product pattern, so integration by parts is appropriate. For a BC-only rational example,

$$ \int \frac{1}{x^2-1},dx, $$

factor the denominator as $(x-1)(x+1)$ and use partial fractions before integrating.

Named misconception — “Any technique will work if I manipulate long enough.” A method can create unnecessary algebra or fail completely when the expression is misclassified. In Skill 1.C, the classification itself is part of the mathematics: identify the form, select the rule, then execute it accurately. Practice should therefore mix techniques rather than group identical integrals together.

Antidifferentiation in applications

Antidifferentiation also converts a rate into a total change. If velocity is $v(t)$, then displacement on $[a,b]$ is

$$ \int_a^b v(t),dt, $$

while distance traveled is

$$ \int_a^b |v(t)|,dt. $$

The absolute value matters because negative velocity indicates motion in the negative direction, not negative distance. Split the interval wherever $v(t)=0$ before evaluating the distance integral.

Worked application. Suppose

$$ v(t)=t-2 $$

for $0\le t\le4$. The particle changes direction at $t=2$. Its distance traveled is

$$ \int_0^2 (2-t),dt+\int_2^4(t-2),dt

2+2=4 $$

units. The signed displacement,

$$ \int_0^4(t-2),dt=0, $$

is different: the particle returns to its starting position.

Washer method and rotation around other axes

When a region is revolved, each thin slice becomes a cross-sectional disk or washer. For slices perpendicular to the axis of rotation,

$$ V=\pi\int_a^b\left(R(x)^2-r(x)^2\right),dx, $$

where $R$ is the outer radius and $r$ is the inner radius. A radius is always a distance to the axis, so revolving around $y=k$ requires expressions such as $|f(x)-k|$, not merely $f(x)$.

For example, revolving the region between $y=x$ and $y=x^2$ on $0\le x\le1$ around the horizontal line $y=2$ produces washers. The outer radius is $R(x)=2-x^2$, and the inner radius is $r(x)=2-x$, giving

$$ V=\pi\int_0^1\left[(2-x^2)^2-(2-x)^2\right],dx. $$

The same principle handles axes other than the coordinate axes: measure every radius from the stated axis, choose slices perpendicular to that axis, and split the integral if the outer and inner boundaries change.

Retrieval check. Classify each expression before integrating: $\int (3x^2+4),dx$, $\int 6x\sqrt{3x^2+1},dx$, and $\int \frac{x}{x^2+1},dx$. Which use basic rules, which use substitution, and why? For an improper integral, what must replace an infinite endpoint?