AP® Calculus BC · Unit 6 · Topic 6.14
Selecting Techniques for Antidifferentiation
Choosing an integration technique means reading the shape of the integrand first: look for a function-and-its-derivative pair, a product of unlike functions, or a rational function to decide between substitution, parts, or partial fractions.
Executing a technique is mechanical once you know which one to use. The hard part is diagnosis: integrals don't announce their method, and picking wrong wastes time on a dead-end calculation before you backtrack. So the real skill in this topic is pattern recognition, trained by looking at thousands of integrands until the shape jumps out.
Scan for three signatures. If you see a chunk of the integrand whose derivative is sitting elsewhere in the expression (like x2 next to x3+1), that screams substitution: let u be the chunk. If you see a product of two genuinely different function types that don't simplify into each other, like a polynomial times arctan(x) or x times ex, that's integration by parts, where you pick u to be the factor that gets simpler when differentiated (arctan(x), ln(x), polynomials) and dv to be the rest. If the integrand is a rational function, a polynomial divided by another polynomial, and the denominator factors, that's partial fractions: split the fraction into simpler pieces first and integrate each one.
This topic sits at the end of the unit because it's the synthesis step: every earlier lesson taught one tool in isolation, and now the integrals stop telling you which tool to grab. The same scanning habit carries forward into using integral tables and into deciding, for improper integrals, whether a closed-form antiderivative is even reachable before you start the algebra.
Key terms
- Substitution
- Rewriting an integral in terms of u, the inner function, when its derivative also appears in the integrand.
- Integration by parts
- A technique for products of unlike functions, based on the formula integral of u dv = uv minus integral of v du.
- Partial fractions
- Splitting a rational function into a sum of simpler fractions with factored denominators before integrating.
- Elementary antiderivative
- An antiderivative expressible using ordinary functions like polynomials, exponentials, logs, and trig.
- Integrand
- The function being integrated, written right after the integral sign.
Watch it explained
- How it worksHow do I scan an integrand for the signature of substitution, parts, or partial fractions?
- Worked exampleFind the integral of x2/(x3 + 1) dx and name the technique that cracked it.
- Worked exampleFind the integral of x arctan(x) dx and justify the method I picked.
- Common mistakeWhy doesn't every integral have an elementary antiderivative at all?
Check yourself
End of topic.