Integral I

$\begin{aligned} I & = \int_0^2 \frac {x+1}{x^2 + x + 2} dx \\ & = \frac 12 \int_0^2 \left(\frac {2x + 1}{x^2+x+2} + \frac 1{x^2+x+2} \right) dx \\ & = \frac {\ln(x^2+x+2)}2 \bigg|_0^2 + \frac 12 \int_0^2 \frac 1{\left(x+\frac 12\right)^2 + \frac 74} dx \\ & = \ln 2 + \color{#3D99F6} \frac 27 \int_0^2 \frac 1{\left(\frac {2x+1}{\sqrt 7}\right)^2 + 1} dx & \small \color{#3D99F6} \text{Let }\tan \theta = \frac {2x+1}{\sqrt 3} \\ & = \ln 2 + \color{#3D99F6} \frac 1{\sqrt 7} \int_{\tan^{-1} \frac 1{\sqrt 7}}^{\tan^{-1} \frac 5{\sqrt 7}} d\theta & \small \color{#3D99F6} \implies \sec^2 \theta \ d\theta = \frac 2{\sqrt 3} dx \\ & = \ln 2 + \frac 1{\sqrt 7} \left(\tan^{-1} \frac 5{\sqrt 7} - \tan^{-1} \frac 1{\sqrt 7} \right) \\ & \approx \boxed{0.966} \end{aligned}$