Integration by the mothed of partial fractions

II- Integration by the mothed of partial fractions
Partial fractions can only be done if the degree of the numerator is strictly less than the degree of the denominator.
For each factor in the denominator we can use the following table to determine the terms we pick up in the partial fraction decomposition.
Factor in denominator Term in partial fraction decomposition
(x+a_1 )(x+a_2 )?(x+a_n ) A_1/((x+a_1 ) )+A_2/((x+a_2 ) )+?+A_n/((x+a_n ) )
Repeated roots: (x+a)^n A_1/((x+a) )+A_2/(x+a)^2 +?+A_n/(x+a)^n
There are several methods for determining the coefficients for each term and we will go over each of those in the following examples.
Example 1: Evaluate the following integrals.
1.??(3x-4)/(x^2-x-6) dx 2.??(x^2-4x+9)/(x+2)^3 dx

1.??(3x-4)/(x^2-x-6) dx
(3x-4)/(x-3)(x+2) =A/((x-3) )+B/((x+2) )
To find A? put x=3 and cover up the factor (x-3)
A=(3x-4)/(x-3)(x+2) =(3×3-4)/((3+2) )=1
To find B : put x=-2 and cover up the factor (x+2)
B=(3x-4)/(x-3)(x+2) =(3×(-2)-4)/((-2-3) )=2
??(3x-4)/(x^2-x-6) dx=??(1/((x-3) )+2/((x+2) )) dx
=ln?|x-3|+2 ln?|x=2|+c
2.??(x^2-4x+9)/(x+2)^3 dx
(x^2-4x+9)/(x+2)^3 =A/(x+2)^3 +B/(x+2)^2 +C/((x+2) )
To find A : multiply both sides by (x+2)^3 and plug in x=-2
x^2-4x+9=A+B(x+2)+C(x+2)^2
(-2)^2-4(-2)+9=A ? A=21
To find B : differentiate once and plug in x=-2
2x-4=B+2C(x+2)
2(-2)-4=B ? B=-8
To find C : differentiate again and plug in x=-2
2=2C ? C=1
??(x^2-4x+9)/(x+2)^3 dx=??(21/(x+2)^3 +(-8)/(x+2)^2 +1/((x+2) )) dx
=(-21)/?2(x+2)?^2 +8/((x+2) )+ln?|x+2|+c
Example 2: Show that
?_0^x?( dx )/([A]_0-x)([B]_0-x) =( 1 )/(([B]_0-[A]_0 ) ) (ln??[A]_0/(([A]_0-x) )-ln??[B]_0/(([B]_0-x) )? ? )
( 1 )/([A]_0-x)([B]_0-x) =( a )/(([A]_0-x) )+( b )/(([B]_0-x) )
x=[A]_0 ? a=( 1 )/([B]_0-[A]_0 )
x=[B]_0 ? b=( 1 )/([A]_0-[B]_0 )
( 1 )/([A]_0-x)([B]_0-x) =( 1 )/([B]_0-[A]_0 )([A]_0-x) +( 1 )/([A]_0-[B]_0 )([B]_0-x)
=( 1 )/(([B]_0-[A]_0 ) ) (( 1 )/(([A]_0-x) )-( 1 )/(([B]_0-x) ))
?_0^x?( dx )/([A]_0-x)([B]_0-x) =( 1 )/(([B]_0-[A]_0 ) ) ?_0^x?(( 1 )/(([A]_0-x) )-( 1 )/(([B]_0-x) ))dx
=? ( 1 )/(([B]_0-[A]_0 ) ) (?-ln???([A]_0-x)+ln?([B]_0-x) ? )?|_0^x
=( 1 )/(([B]_0-[A]_0 ) ) (?-ln???([A]_0-x)+ln??([B]_0-x)+ln??[A]_0-ln??[B]_0 ? ? ? ? )
=( 1 )/(([B]_0-[A]_0 ) ) (ln??[A]_0/(([A]_0-x) )-ln??[B]_0/(([B]_0-x) )? ? )


Evaluate the following integral
1.??(3x+5)/(x^2-2x-15) dx 2.??(?2x?^2+3x+2)/(x+1)^3 dx