Topic 6.14 Selecting Techniques for Antidifferentiation: choose between rewriting, basic rules, and substitution to evaluate an integral.

What this topic is asking

The College Board (Topic 6.14) is the strategy topic for integration: given an integral, decide whether to rewrite algebraically, apply a basic rule directly, or use u-substitution. On the AB exam the antidifferentiation toolkit is small, so the skill is matching the integrand to the right tool quickly.

The decision sequence

A worked technique selection

Rewriting beats substitution when it can

A common inefficiency is reaching for substitution when simple algebra would do. The integral $\int \frac{x^3 + 2x}{x}\,dx$ tempts a substitution, but splitting it to $\int (x^2 + 2)\,dx = \frac{x^3}{3} + 2x + C$ is faster and less error-prone. Similarly $\int (x + 1)^2\,dx$ can be expanded to $\int (x^2 + 2x + 1)\,dx$ rather than substituted. The exam deliberately includes integrals where both routes work, rewarding students who simplify first. Reach for substitution only when the integrand genuinely has a composite with its inside-derivative present.

Knowing the boundary of the AB toolkit

Part of selecting a technique is recognizing what is out of scope. AP Calculus AB does not assess integration by parts, partial fractions, or trigonometric substitution; those are BC topics. So if an AB integral seems to need one of those, you have almost certainly missed an algebraic simplification or a substitution. This boundary is a useful constraint: when stuck, look again for a rewrite or a $u$-substitution rather than assuming an advanced method is required. Every antiderivative the AB exam asks for is reachable with rewriting, the basic rules, and substitution.

Diagnostic questions for choosing a method

A few quick questions sort most integrals. Is the integrand a single basic function (a power, $e^x$, $\sin$, $\cos$, $\frac{1}{x}$)? Then apply the basic rule. Is it a fraction over a monomial, or a small product, or a root or reciprocal? Then rewrite first. Does it contain a composite function alongside (a constant times) the derivative of the inside? Then substitute. Asking these in order, basic rule, rewrite, substitute, resolves the choice without trial and error. The diagnostic also flags when none apply on the AB scale, which signals that a rewrite has been missed, since the exam never poses an AB integral that genuinely needs a BC technique.

Building speed on the no-calculator section

Because integration appears heavily on the no-calculator parts, speed comes from instant recognition rather than deliberation. With practice, $\int \frac{x}{x^2+1}\,dx$ reads as a logarithm at a glance, $\int (3x-1)^4\,dx$ as a quick linear substitution, and $\int \frac{x^2+x}{x}\,dx$ as a rewrite into $x + 1$. The goal is to spend your time on the genuinely substantive integrals, not on deciding how to start the routine ones. Drilling the three approaches until the right one is obvious from the form of the integrand is what frees time and reduces errors under the timed conditions of the exam.