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For every function \( f(x) \) which is twice differentiable, these will be good approximation of
\[
\int_{a}^{b} f(x) d x=\left(\frac{b-a}{2}\right)\{f(a)+f(b)\},
\]
P
WV
for more acurate results for \( c \in(a, b) \),
\[
F(c)=\frac{c-a}{2}[f(a)-f(c)]+\frac{b-c}{2}[f(b)-f(c)]
\]
When \( c=\frac{a+b}{2} \)
\[
\int_{a}^{b} f(x) d x=\frac{b-a}{4}\{f(a)+f(b)+2 f(c)\} d x \quad(2006,6 \mathrm{M})
\]
Good approximation of \( \int_{0}^{\pi / 2} \sin x d x \), is
(a) \( \pi / 4 \)
(b) \( \pi(\sqrt{2}+1) / 4 \)
(c) \( \pi(\sqrt{2}+1) / 8 \)
(d) \( \frac{\pi}{8} \)
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