(a) \(\int _{0} ^{1} \frac{3}{1 + x^{2}} \mathrm {d} x\)

\(x\)	0	0.25	0.50	0.75	1.0
\(x^{2}\)	0	0.06	0.25	0.56	1.0
\(1 + x^{2}\)	1	1.06	1.25	1.56	2.0
\(\frac{3}{1 + x^{2}}\)	3	2.82	2.40	1.92	1.5

h = 0.25 ;
\(y_{1} = 3, y_{2} = 2.82 , y_{3} = 2.40 , y_{4} = 1.92 , y_{5} = 1.5\)
\(y_{1} + y_{5} = 3 + 1.5 = 4.5\)
\(y_{2} + y_{3} + y_{4} = 2.82 + 2.40 + 1.92 = 7.14\)
\(2(y_{2} + y_{3} + y_{4}) = 14.28\)
= \(3[\frac{1}{2} h ( y_{1} + y_{5} + 2(y_{2} + y_{3} + y_{4})]\)
= \(3[\frac{1}{2} (0.25) (4.5 + 14.28)]\)
= \(3(0.125)(18.78) = 2.3475\)
\(\approxeq 2.348\) (4 sig. figures).
(b) \(A = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix}\)
\(A^{2} + A + 2I = \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} + \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} + 2 \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}\)
= \(\begin{pmatrix} 4 + 3 & 2 + 2 \\ 6 + 6 & 3 + 4 \end{pmatrix} + \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} + \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\)
= \(\begin{pmatrix} 7 & 4 \\ 12 & 7 \end{pmatrix} + \begin{pmatrix} 2 & 1 \\ 3 & 2 \end{pmatrix} + \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix}\)
= \(\begin{pmatrix} 11 & 5 \\ 15 & 11\end{pmatrix}\)
\(T : \begin{pmatrix} 1 \\ 2 \end{pmatrix} \to \begin{pmatrix} 11 & 5 \\ 15 & 11 \end{pmatrix} \begin{pmatrix} 1 \\ 2 \end{pmatrix}\)
=\(\begin{pmatrix} 11 + 10 \\ 15 + 22 \end{pmatrix} = \begin{pmatrix} 21 \\ 37 \end{pmatrix}\)
The image of (1, 2) = (21, 37).