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6.14 Selecting Techniques for Antidifferentiation

Syllabus
2020
Topic
6.14
Level
—

Let the Integrand Choose the Technique

Before integrating, simplify and classify the integrand. Ask which recognizable structure is present and choose a method that converts the expression into a more familiar antiderivative; a good choice reduces complexity rather than merely changing notation.

Visible structure First method to test Diagnostic question
direct sum, constant multiple, or known derivative pattern basic antiderivative rule can terms be integrated immediately?
composite function with an inner derivative factor substitution is f′(x)f'(x) present with g(f(x))g(f(x))?
improper rational function or useful quadratic rewrite algebra first will division or completing the square reveal a known form?
product where one factor simplifies when differentiated integration by parts will the new integral be simpler?
proper rational function with distinct linear factors linear partial fractions can it split into logarithmic pieces?

Examples of selection: ∫2xcos⁡(x2) dx\int 2x\cos(x^2)\,dx suggests u=x2u=x^2 because du=2x dxdu=2x\,dx is present. In contrast, ∫xex dx\int xe^x\,dx has no inner-function pattern, but differentiating xx simplifies it, so integration by parts is appropriate.

For ∫x2+1x+1 dx\int\frac{x^2+1}{x+1}\,dx, the numerator degree signals long division before integrating. For ∫1(x+1)(x+2) dx\int\frac{1}{(x+1)(x+2)}\,dx, the proper rational form with distinct linear factors signals partial fractions. These decisions come from structure, not from the Topic number.

No single acronym replaces inspection, and the first attempted method is not automatically correct. Check that each transformation is equivalent on the original domain, retain definite-integral bounds correctly, and differentiate an indefinite result to verify it. If the new integral is harder, reconsider the choice.