Typical integrals:
Example | Box |
Substitution
Here | |
Integration by parts | |
Integration by parts | |
Substitution
Here It also fits the box "trigonometric integrals", but there the recommendation there is exactly the substitution as above. | |
Rational function (partial fractions decomposition) | |
Rational function (partial fractions decomposition)
Better: Substitution (numerator is the derivative of denominator). | |
Integration by parts | |
Substitution
Here | |
Rational function (partial fractions decomposition) | |
Integration by parts | |
Integration by parts | |
Trigonometric integral
Either the universal substitution | |
Integral with a root of a quadratic
General method: indirect substitution Here we have a simpler alternative: Factoring out the constant leads to an elementary integral. | |
Rational function
Completing the square and factoring out a constant leads to an elementary integral |
A well-prepared student should make such associations on the fly. Note that although the list is by no means exhaustive, experience suggests that if a student is really familiar with integrals of the listed types, he should do well on most integration problems.