Question

integration of (2x+1)√x²+x+1

Answer 1

let

x^2 + x +1 = t

(2x + 1 )dx = dt

integrate

√t

Answer 2

$\int{} (2x + 1) \sqrt{ (x {}^{2} + x + 1)} \: dx\\ \\ Let \: \: \: t \: = (x {}^{2} + x + 1) \\ dt \: = \: (2x + 1) \: dx \\ \\ \int{} \sqrt{t} \: dt \: = \frac{2}{3} \: {t}^{ \frac{3}{2} } + \: C \\ \\ \\ \large \boxed{ \int{} (2x + 1) \sqrt{ (x {}^{2} + x + 1)} \: dx \: = \frac{2}{3} {( {x}^{2} + x + 1) }^{ \frac{3}{2} } + \: C}$