257
edits
Changes
Integration by parts (view source)
Revision as of 17:44, December 31, 2008
, 17:44, December 31, 2008→Rapid Repeated Integration: added further explanation
==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.
===Example 3===
For example, integrating:
For example, integrating:
:<big><math>\int x^4 sin(x)\,dx</math></big>
:<big><math>\int x^4 sin(x)\,dx</math></big>
|-
|-
| <math>x^4</math>
| <math>x^4</math>
| <math>sin(x)</math>
| <math>\sin(x)</math>
|-
|-
| <math>4x^3</math>
| <math>4x^3</math>
| <math>-cos(x)</math>
| <math>-\cos(x)</math>
|-
|-
| <math>12x^2</math>
| <math>12x^2</math>
| <math>-sin(x)</math>
| <math>-\sin(x)</math>
|-
|-
| <math>24x</math>
| <math>24x</math>
| <math>cos(x)</math>
| <math>\cos(x)</math>
|-
|-
| <math>24</math>
| <math>24</math>
| <math>sin(x)</math>
| <math>\sin(x)</math>
|-
|-
| <math>0</math>
| <math>0</math>
| <math>-cos(x)</math>
| <math>-\cos(x)</math>
|}
|}
:<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><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>=<big><math>\ -x^4cos(x)+4x^3sin(x)+12x^2cos(x)-24xsin(x)-24cos(x)+C</math></big>
While this method is very useful when part of the integrand becomes zero through [[derivative|derivation]], it does not work in cases such as Example 2 above.
== See Also ==
== See Also ==