Typical integrals:

Example | Box |

Substitution
Here y = x^{5} − x. | |

Integration by parts | |

Integration by parts | |

Substitution
Here y = sin(x).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 y = ln(x). | |

Rational function (partial fractions decomposition) | |

Integration by parts | |

Integration by parts | |

Trigonometric integral
Either the universal substitution y = tan(x/2),y = tan(x). | |

Integral with a root of a quadratic
General method: indirect substitution x = 4sin(t).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.