Techniques of Integration
Finding antiderivatives of elementary functions is often considerably harder than finding their derivatives. For some elementary functions, it is impossible to find an antiderivative in terms of other elementary functions. See the article on elementary functions for further information.
We have various methods at our disposal:
- the linearity of integration allows us to break complicated integrals into simpler ones
- integration by substitution, often combined with trigonometric identities or the natural logarithm
- integration by parts to integrate products of functions
- the inverse chain rule method, a special case of integration by substitution
- the method of partial fractions in integration allows us to integrate all rational functions (fractions of two polynomials)
- the Risch algorithm
- integrals can also be looked up in a table of integrals
- when integrating multiple times, we can use certain additional techniques, see for instance double integrals and polar coordinates, the Jacobian and the Stokes' theorem
- computer algebra systems can be used to automate some or all of the work involved in the symbolic techniques above, which is particularly useful when the algebraic manipulations involved are very complex or lengthy
- if a function has no elementary antiderivative (for instance, exp(-x2)), its definite integral can be approximated using numerical integration
- to calculate the ( times) repeated antiderivative of a function Cauchy's formula is useful (cf. Cauchy formula for repeated integration):
Read more about this topic: Antiderivative
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