Difference between revisions of "Methods of integration"

Revision as of 19:15, August 20, 2010

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-This article details several methods of [[integral|integration]] for advanced high school or early university student, each with an example.
-==Estimation==
-There are various methods of estimating integrals. This is particularly useful if the rule for integrating is unknown or extremely complicated, as in the case of <math>e^{x^2}</math>. The primary method of estimation is the Riemann integral.
-<br />Another major method, known as [[Simpson's rule]] uses the Riemann integral. Instead of using linear estimations, Simpson's rule allows for the use of [[parabola]]s or other higher order expressions to estimate integrals.
-:{{main|Simpson's rule}}
-==Integration by Parts==
-{{main|Integration by parts}}
-'''Integration by parts''' is a special 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]].
-The rule for '''integration by parts''' is stated as follows:
-:<big><math>\int f(x) g'(x)\,dx = f(x) g(x) - \int f'(x) g(x)\,dx,</math></big>
-:or
-:<big><math>\int u\,dv = uv - \int v\,du\,</math></big>
-This rule is often useful when one function is a power of ''x'' and the other function  is a trigonometric function or ''e'' raised to a 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==
-{{Main|Partial fractions in integration}}
-'''Integration by partial fractions''' is a technique to facilitate the integration of a rational expression by partial fraction decomposition.
-Given an integral
-<br />
-:<math>\int \frac {f(x)}{g(x)}dx</math>
-<br />
-where <math>f(x)</math> and <math>g(x)</math> are both polynomials, integration by partial fractions shows how to separate the problem into multiple integrals before integrating.
-==Algebraic Substitution==
-'''Integration by Algebraic Substitution''' is a technique to facilitate the integration of a rational expression by substituting a more complicated expression with a variable.
-Given an integral
-:<big><math>\int\frac{2x}{x^2+3}dx</math></big>
-We can substitute the term :<big><math>x^2+3</math></big> with a u.  Giving us
-:<big><math>u=x^2+3</math></big>
-We then take the derivative of u with respect to x,
-:<big><math>\frac{du}{dx}=2x</math></big>
-We then set the terms equal to du,
-:<big><math>du=2xdx</math></big>
-Now we are ready to rewrite the integral,
-:<big><math>\int\frac{2x}{x^2+3}dx=\int\frac{1}{u}du</math></big>
-We can rewrite the integral this way due to the subsitution of the x terms with the u terms.
-Now we can solve the integral in terms of u.
-:<big><math>\int\frac{1}{u}du=ln|u|+c</math></big>
-Now we replace u with the term :<big><math>x^2+3</math></big> to get,
-:<big><math>ln|x^2+3|+c</math></big>
-We can check this by taking the derivative of :<big><math>ln|x^2+3|</math></big>,
-:<big><math>\frac{d}{dx}ln|x^2+3|=\left (\frac{1}{x^2+3} \right)(2x)=\frac{2x}{x^2+3}</math></big>
-==Trigonometric Substitution==
-'''Integration by Trigonometric Substitution''' is a technique to facilitate the integration of a rational expression by substituting a more complicated radical expression with a trigonometric expression.
-Given an integral
-:<big><math>\int\frac{1}{\sqrt{9-x^2}}dx</math></big>
-By looking at the radical we can determine that it represents the base of a right triangle by understanding the [[Pythagorean theorem]].
-:<big><math>\sqrt{9-x^2}=3+x</math></big> where 3 is the hypotenuse and x is the height of the triangle.
-This allows us to rewrite the expression to :<big><math>sin\theta=\frac{x}{3}</math></big>.  This allows us to substitute x with :<big><math>3sin\theta</math></big>.
-Now to do the substitution
-:<big><math>9-x^2=9-(3sin\theta )^2</math></big>
-:<big><math>9-(3sin\theta )^2=9-9sin^2\theta </math></big>
-:<big><math>9-9sin^2\theta=9(1- sin^2\theta) </math></big>
-And by use of trigonometric identities we know that
-:<big><math>1- sin^2\theta=cos^2\theta </math></big>
-:<big><math>9(1- sin^2\theta)=9cos^2\theta </math></big>
-Therefore
-:<big><math>\sqrt{9-x^2}=3cos\theta </math></big>
-We are not done yet, we must also take the derivative of :<big><math>3sin\theta</math></big>
-:<big><math>\frac{dx}{d\theta}3sin\theta=3cos\theta </math></big>
-By partial derivatives we move the :<big><math>{d\theta}</math></big> over.
-:<big><math>dx=3cos\theta d\theta </math></big>
-Now we are ready to rewrite our integral.
-:<big><math>\int\frac{1}{\sqrt{9-x^2}}dx=\int\frac{3cos\theta}{3cos\theta}d\theta=\int d\theta=\theta+c</math></big>
-From our trigonometric expression :<big><math>x=3sin\theta</math></big> we can see that
-:<big><math>\theta=sin^{-1}\left(\frac{x}{3}\right)+c</math></big> giving us the final solution.
-:<big><math>\int\frac{1}{\sqrt{9-x^2}}dx=sin^{-1}\left(\frac{x}{3}\right)+c </math></big>
-== Other methods of integration ==
-The primary methods of integration include:
-* [[Integration by parts]]
-* [[Integration using polar coordinates]]
-* [[Multiple integration]]
-* [[Residue calculus]] (for definite integrals)
-* [[Method of simultaneous convolutions]]
-* [[Mellin transform]]s
-* [[Inflation-restriction sequence]]s
-==See also==
-*[[Fundamental Theorem of Calculus]]
-*[[Calculus]]
-*[[Integration]]
-==External links==
-*[http://user.mendelu.cz/marik/maw/index.php?lang=en&form=integral Antiderivative Solver]
-[[Category:calculus]]
-[[Category:integration]]

Difference between revisions of "Methods of integration"

Revision as of 19:15, August 20, 2010

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