Question

Integrate the following with respect to x.

`(4x + 2) sqrt(x^2 + x + 1)`

Answer 1

Let f(x) = x² + x + 1

Then f'(x) = 2x + 1

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

= `2int "f'"(x) sqrt("f"(x))  "d"x`

= `2 int ["f"(x)]^(1/2) "f'"(x)  "d"x`

= `2["f"(x)]^(3/2)/3 + "c"`

= `4/3(x^2 + x + 1)^(3/2) + "c"`