<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 taylor series you put in the comment is wrong. You should get 
$$
\ln\left(1+\frac{x^2}{4}\right)= \frac{x^2}{4}-\frac{x^4}{32}+\frac{x^6}{192}-\frac{x^8}{1024}+O(x^{10})
$$
Integrating up to the 2nd power you get 
$$
\int_0^{1/2} \frac{x^2}{4}dx = \frac{1}{96} \simeq 0.010417
$$
Integrating up to the 4th power you get 
$$
\int_0^{1/2} \frac{x^2}{4}-\frac{x^4}{32} dx = \frac{ 157}{15360 } =0.01022135416
$$
Meanwhile the real result is 
$$
\int_0^{1/2} \ln\left(1+\frac{x^2}{4}\right) dx = \frac12 \left(-2+\ln(17/16)+8 \tan^{-1}(1/4)\right) \simeq 0.010227
$$
Since when we integrate up to the 4rth power we left out the terms with order $O(x^{2n})$ for $n> 2$, we get a result with error of order less than $10^{-4}$, indeed
$$
0.010227-0.01022135416=  0.00000564584< 0.0001= 10^{-4}
$$
</s>