Section F.1 A few general tactics
In trying to evaluate an integral with the collection of methods we have seen in Chapters 1 and 3, some general ideas are worth keeping in mind:
- Find antiderivatives first
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It is often best to start by seeking any one antiderivative \(F(x)\) for the function involved and deal later with the constant of integration in an indefinite integral and the limits of a definite integral.However when using substitutions on a definite integral, you can avoid substituting back to the original variable by instead changing the limits of integration to the corresponding values for the new variable.
- Start with the easiest possibilities
- Several techniques might apply to one integral, so start try the easiest first (recognizing a previously known integral, from memory or tables) and work through to the more sophisticated options (like inverse trigonometric substitutions and integration by parts.)
- Repeat as necessary
- All methods except recognizing a known integral give one or several new simpler integrals, so the process needs to be applied repeatedly until every part of the answer is found by reducing to a previously known integral.
