Evaluate: (i) `int((x^(3) + 4x^(2) -3x -2))/((x+2)) dx` , (ii) `int((x^(4) +1)/(x^(2) +1))dx`

Explanation

(i) On dividing `(x^(3) + 4x^(2) - 3x -2)` by `(x+2)`, we get
`int((x^(3) + 4x^(2) -3x-2))/((x+2)) dx = int{x^(2) + 2x-7 + 12/(x+2)} dx`
`= intx^(2)dx + 2intxdx -7 intdx + 12 int1/(x+2) dx`
`= (x^(3))/(3) + 2.(x^(2))/(2) - 7x+12 log |x+2| + C`
`= (x^(3))/(3) + x^(2) - 7x + 12 log |x+2| + C`.
(ii) On dividing `(x^(4) +1)` by `(x^(2) + 1)`, we get
`int((x^(4) + 1)/(x^(2) +1)) dx = int[x^(2) - 1 +(2)/((x^(2) + 1)) ]dx`
`= intx^(2)dx - intdx + 2 int(1)/(x^(2) + 1)dx`
`= (x^(3))/(3) - x +2 tan^(-1)x + C`.