Calculus
posted by Maryam .
1. Evaluate the following integrals
(a) 4x2 +6x−12 / x3 − 4x dx

I will assume you meant
(4x^2 +6x−12)/(x^3 − 4x)
= (4x^2 + 6x  12)/( x(x+2)(x2) )
split it into partial fractions
let
A/x + B/(x+2) + C/(x2) = (4x^2 + 6x  12)/( x(x+2)(x2) )
multiply by x(x+2)x2)
A(x^24) + Bx(x2) + Cx(x+2) = 4x^2 + 6x  12
This must be true for all x's
let x = 0
4A = 12
A=3
let x = 2
8C = 16 + 1212 = 16
C = 2
let x = 2
8B = 16 + 1212 = 8
B = 1
so ∫ (4x^2 + 6x  12)/( x(x+2)(x2) ) dx
= ∫3/x dx + ∫1/(x+2)dx + ∫2/(x2) dx
= 3ln x  ln(x+2) + 2ln(x2) + k, where k is a constant