Write out the form of the partial fraction decomposition of the function appearing in the integral: integrate of (2x-44)/(x^2+6x-16)dx

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jeew-m | College Teacher | (Level 1) Educator Emeritus

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`x^2+6x-16`

`= x^2+8x-2x-16`

`= x(x+8)-2(x-8)`

= (x+8)(x-2)

 

`(2x-44)/(x^2+6x-16) = A/(x+8)+B/(x-2)`

                     ` 2x-44 = A(x-2)+B(x+8)`

Comparing components;

x-----> `2 = A+B` ----(1)

constant---> `-44 = -2A+8B` ----(2)

 

By solving (1) and (2) you will get;

`A = 6`

`B = -4`

 

`int(2x-44)/(x^2+6x-16)dx`

= int [6/(x+8)-4/(x-2)]dx

`= int 6/(x+8)dx-int 4/(x-2)dx`

`= 6ln(x+8)-4ln(x-2)+C` where C is a constant

`= ln((x+8)^6/(x-2)^4)+C`

 

`int(2x-44)/(x^2+6x-16)dx = ln((x+8)^6/(x-2)^4)+C`

 

 

 

 

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