Question

Integrate the following with respect to x.

`sqrt(1 + x + x^2)`

Answer 1

`int sqrt(1 + x + x^2)  "d"x = int sqrt((x + 1/2)^2 - 1/4 + 1)  "d"x`

= `int sqrt((x + 1/2)^2 + 3/4)   "d"x`

= `int sqrt((x + 1/2)^2 + (sqrt(3)/2)^2)  "d"x`

= `((x + 1/2))/2 sqrt(1 + x + x^2) + 3/8  log|(x + 1/2) + sqrt(1 + x + x^2)| + "c"`

{Using the formula for `int sqrt(x2 + "a"^2)  "d"x` with 'x' as `x + 1/2` and 'a' as `sqrt(3)/2`}