Write the partial fraction equation:
(x^2+4x+12)/((x+2)(x^2+4)) = A/(x+2) + (Bx)/(x^2+4) + C/(x^2+1)
Multiply both sides by (x+2)(x^2+4):
x^2+4x+12 = A(x^2+4) + (Bx)(x+2) + C(x+2)" [1]"
Make B and C disappear by letting x = -2
(-2)^2+4(-2)+12 = A((-2)^2+4)
8 = A(8)
A = 1
Substitute the value for A into equation [1]:
x^2+4x+12 = (x^2+4) + (Bx)(x+2) + C(x+2)" [2]"
Make B disappear by letting x = 0
0^2+4(0)+12 = (0^2+4) + C(0+2)
8 =2C
C = 4
Substitute the value for C into equation [2]:
x^2+4x+12 = (x^2+4) + (Bx)(x+2) + 4(x+2)" [3]"
Let (x = 1):
1^2+4(1)+12 = (1^2+4) + (B(1))((1)+2) + 4(1+2)
0 = 3B
B = 0
Remove the term for B from equation [3]
x^2+4x+12 = (x^2+4) + 4(x+2)
Divide by (x+2)(x^2+4):
(x^2+4x+12)/((x+2)(x^2+4)) = 1/(x+2) + 4/(x^2+4)" [4]"
Equation [4] gives us the template for the integrals:
int(x^2+4x+12)/((x+2)(x^2+4))dx = int1/(x+2)dx + 4int1/(x^2+1)dx
Both integrals are well known:
int(x^2+4x+12)/((x+2)(x^2+4))dx = ln|x+2| + 4tan^-1(x)+ C
