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Question
Integrate the following with respect to x.
`(4x + 2) sqrt(x^2 + x + 1)`
Solution
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"`
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