<s>[INST] Approximation of integration I want to estimate the integral $\int_0^{1/2}\ln(1+ \frac{x^2}{4})$
with error at most $10^{-4}$.
Any help will be appreciated.

I have calculated the power series of $\ln(1+ \frac{x^2}{4})$ which is
$$\sum_{n=0}^{n= \infty} \frac{(-1)^n}{(2n+2)\cdot 2^{2n+1}}x^{2n+2},$$
and radius of convergence $= 1/2$. Now calculate
$$\int_0^{1/2}\sum_{n=0}^{n= \infty} \frac{(-1)^n}{(2n+1)\cdot 2^{2n+1}}x^{2n+1}\,dx =  \sum_{n=0}^{n= \infty} \frac{(-1)^n}{(2n+3)(2n+2)\cdot 2^{2n+1}}(\frac{1}{2})^{2n+3}.$$
 [/INST] The integral can be computed analytically:
$x \ln \left(\frac{1}{4} \left(x^2+4\right)\right)-2 x+4 \tan ^{-1}\left(\frac{x}{2}\right)$
and with the limits becomes 
$\frac{1}{2} \left(-2+\ln \left(\frac{17}{16}\right)+8 \tan ^{-1}\left(\frac{1}{4}\right)\right)$
or
$0.010227$.
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