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Methods of integration (view source)
Revision as of 03:30, December 30, 2008
, 03:30, December 30, 2008changed headings to allow for easier navigation
This article details several methods of [[integral|intergration]] for advanced high school or early university student, each with an example
This article details several methods of [[integral|intergration]] for advanced high school or early university student, each with an example
==Integration by Parts==
{{main|Integration by parts}}
{{main|Integration by parts}}
'''Integration by parts''' is a special [[Techniques of integration|technique]] to facilitate the integration of the product of two functions that otherwise lack an obvious integral. This technique can be proven with the [[product rule]].
'''Integration by parts''' is a special [[Techniques of integration|technique]] to facilitate the integration of the product of two functions that otherwise lack an obvious integral. This technique can be proven with the [[product rule]].
Note that it may be necessary to repeat the '''integration by parts''' several times, one for each power of ''x''.
Note that it may be necessary to repeat the '''integration by parts''' several times, one for each power of ''x''.
==Partial Fractions==
'''Integration by partial fractions''' is a [[Techniques of integration|technique]] to facilitate the integration of a rational expression by partial fraction decomposition.
'''Integration by partial fractions''' is a [[Techniques of integration|technique]] to facilitate the integration of a rational expression by partial fraction decomposition.
:<big><math>\int\frac{3x+11}{x^2-x-6}dx=4ln|x-3|-ln|x+2|+c</math></big>
:<big><math>\int\frac{3x+11}{x^2-x-6}dx=4ln|x-3|-ln|x+2|+c</math></big>
==Algebraic Substitution==
'''Integration by Algebraic Substitution''' is a [[Techniques of integration|technique]] to facilitate the integration of a rational expression by substituting a more complicated expression with a variable.
'''Integration by Algebraic Substitution''' is a [[Techniques of integration|technique]] to facilitate the integration of a rational expression by substituting a more complicated expression with a variable.
:<big><math>\frac{d}{dx}ln|x^2+3|=(\frac{1}{x^2+3})(2x)=\frac{2x}{x^2+3}</math></big>
:<big><math>\frac{d}{dx}ln|x^2+3|=(\frac{1}{x^2+3})(2x)=\frac{2x}{x^2+3}</math></big>
==Trigonometric Substitution==
'''Integration by Trigonometric Substitution''' is a [[Techniques of integration|technique]] to facilitate the integration of a rational expression by substituting a more complicated radical expression with a trigonometric expression.
'''Integration by Trigonometric Substitution''' is a [[Techniques of integration|technique]] to facilitate the integration of a rational expression by substituting a more complicated radical expression with a trigonometric expression.