Difference between revisions of "Methods of integration"

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====Rapid Repeated Integration====
 
====Rapid Repeated Integration====
 
Rapid Repeated Integration is a shortcut method for reduction problems that require Integration by Parts. It is especially useful when one function's derivative reduces to zero.
 
Rapid Repeated Integration is a shortcut method for reduction problems that require Integration by Parts. It is especially useful when one function's derivative reduces to zero.
For Example, lets say we want to integrate:
+
For example, integrating:
 
:<big><math>\int x^4 sin(x)\,dx</math></big>
 
:<big><math>\int x^4 sin(x)\,dx</math></big>
First, we start off by making a table of x^4 and sin(x). On the first column, we keep taking the derivative of x^4 until it reaches zero. On the second column, we keep integrating sin(x):
+
We start off by making a table of <math>x^4</math> and <math>sin(x)</math>. On the first column, we keep taking the derivative of <math>x^4</math> until it reaches zero. On the second column, we keep integrating sin(x):
  
 
{| class="wikitable"
 
{| class="wikitable"
 
|-
 
|-
! x^4
+
! <math>f(x)</math>
! sin(x)
+
! <math>g(x)</math>
 
|-
 
|-
| 4x^3
+
| <math>x^4</math>
| -cos(x)
+
| <math>sin(x)</math>
 
|-
 
|-
| 12x^2
+
| <math>4x^3</math>
| -sin(x)
+
| <math>-cos(x)</math>
 
|-
 
|-
| 24x
+
| <math>12x^2</math>
| cos(x)
+
| <math>-sin(x)</math>
 
|-
 
|-
| 24
+
| <math>24x</math>
| sin(x)
+
| <math>cos(x)</math>
 
|-
 
|-
| 0
+
| <math>24</math>
| -cos(x)
+
| <math>sin(x)</math>
 +
|-
 +
| <math>0</math>
 +
| <math>-cos(x)</math>
 
|}
 
|}
  
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:<big><math>\int x^4 sin(x)\,dx</math></big> = <big><math>\ -x^4cos(x)-(-4x^3sin(x))+(12x^4cos(x))-(-24xsin(x))+(24cos(x))</math></big>=<big><math>\ -x^4cos(x)+4x^3sin(x)+12x^2cos(x)+24xsin(x)-24cos(x)+c</math></big>
 
:<big><math>\int x^4 sin(x)\,dx</math></big> = <big><math>\ -x^4cos(x)-(-4x^3sin(x))+(12x^4cos(x))-(-24xsin(x))+(24cos(x))</math></big>=<big><math>\ -x^4cos(x)+4x^3sin(x)+12x^2cos(x)+24xsin(x)-24cos(x)+c</math></big>
 +
 
===Partial Fractions===
 
===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.

Revision as of 02:41, December 30, 2008

This article details several methods of intergration for advanced high school or early university student, each with an example

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 utilizes the insight of the product rule.

The rule for integration by parts is stated as follows:

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.

Rapid Repeated Integration

Rapid Repeated Integration is a shortcut method for reduction problems that require Integration by Parts. It is especially useful when one function's derivative reduces to zero. For example, integrating:

We start off by making a table of and . On the first column, we keep taking the derivative of until it reaches zero. On the second column, we keep integrating sin(x):

Then, we cross multiply diagonally from the left to the right and we add the product to the product of the next product, alternating signs as we go (remember to start with a minus). So then the integral becomes:

= =

Partial Fractions

Integration by partial fractions is a technique to facilitate the integration of a rational expression by partial fraction decomposition.

Given an integral

The first step is to factor the denominator as much as possible and get the form of the partial fraction decomposition. Doing this gives,

This allows us to split the fraction in to sums by cross multiplying the denominators,

Therefore we can restate the problem,

Now we can solve for A and B by subsituting x with a value that allows the term to go to 0. For example,

We let :,

We let :,

We then plug in the values of A and B and get,

Now we can solve the integral.

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

We can substitute the term : with a u. Giving us

We then take the derivative of u with respect to x,

We then set the terms equal to du,

Now we are ready to rewrite the integral,

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.

Now we replace u with the term : to get,

We can check this by taking the derivative of :,

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

By looking at the radical we can determine that it represents the base of a right triangle by understanding the Pythagorean theorem.

where 3 is the hypotenuse and x is the height of the triangle.

This allows us to rewrite the expression to :. This allows us to substitute x with :. Now to do the substitution

And by use of trigonometric identities we know that

Therefore

We are not done yet, we must also take the derivative of :

By partial derivatives we move the : over.

Now we are ready to rewrite our integral.

From our trigonometric expression : we can see that

giving us the final solution.