Changes

====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>0</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>=<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.