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Methods of integration (view source)
Revision as of 20:30, October 26, 2008
, 20:30, October 26, 2008no edit summary
The rule for '''integration by parts''' is stated as follows:
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>
:<big><math>\int f(x) g'(x)\,dx = f(x) g(x) - \int f'(x) g(x)\,dx,</math></big>
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''.
====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, lets say we want to integrate:
:<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):
{| class="wikitable"
|-
! x^4
! sin(x)
|-
| 4x^3
| -cos(x)
|-
| 12x^2
| -sin(x)
|-
| 24x
| cos(x)
|-
| 24
| sin(x)
|-
| 0
| -cos(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:
:<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.