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\begin{document}
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% !!! IMAGES START HERE !!!

{\newpage\clearpage
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\fbox{MM 246 -- Numerical Methods }%
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$ J = J_{T,n} + \mathcal{E}_{T,n}$%
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$\displaystyle J_{T,n} = \frac{h}{2} \left(f_0 + 2 \sum_{i=1}^{n-1} f_i + f_n \right),$%
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$\displaystyle \mathcal{E}_{T,n} = -  \frac{h^2}{12} (b-a) f''(\tau), \qquad \tau \in [a,b].$%
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$ n=2m$%
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$ J = J_{S,n} + \mathcal{E}_{S,n}$%
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$\displaystyle J_{S,n} = \frac{h}{3} \left(f_0 + 
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2 \sum_{i=1}^{n/2-1} f_{2i} 
+ f_n \right),$%
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$\displaystyle \int_a^b f(x) \mathrm{d}x= \int_a^b \frac{x}{1 + x^2} \mathrm{d}x.$%
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$ a$%
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$ b$%
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$ n$%
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$ \int_0^2 \frac{x}{1 + x^2} \mathrm{d}x$%
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$ n=4$%
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$ n=8$%
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$ n=16$%
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$\displaystyle \frac{1}{2} \ln(5) =0.8047189560$%
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$ n = 4, 8, 16$%
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$ J_{T,n}$%
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$\displaystyle \int_0^2 x \mathrm{d}x, \quad 
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\int_0^2 x^3 \mathrm{d}x, \quad
\int_0^2 \exp(x) \mathrm{d}x.$%
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$ \Bigl| \int_0^2 x \mathrm{d}x- J_{T,n} \Bigr|$%
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$ \Bigl| \int_0^2 x^2 \mathrm{d}x- J_{T,n} \Bigr|$%
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$ \Bigl| \int_0^2 \exp(x) \mathrm{d}x- J_{T,n} \Bigr|$%
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$ \int_0^2 \exp(x) \mathrm{d}x$%
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$ |J - J_{T,n}| \leq 10^{-5}$%
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\end{document}