User:Superdan006/Sandbox/hw2


 * $$\int_0^1 \frac{4}{1+x^2}\,dx \ $$


 * $$\int_a^b f(x)\,dx \approx\frac{b-a}{2n}\sum^{n-1}_{i=0}[f_i+f_{i+1}] $$


 * $$R(n,m)=R(n,m-1)+\frac{1}{4^m-1}[R(n,m-1)-R(n-1,m-1)] $$


 * $$\int_a^b f(x)\,dx \approx\sum_i\omega_if(x_i) $$


 * $$\int_0^{2\pi} \frac{\cos{2x}}{e^x}\,dx \ $$


 * $$ \int_{a}^{b} f(x) \, dx \approx \frac{(b-a)}{6}\left[f(a) + 4f\left(\frac{a+b}{2}\right)+f(b)\right]$$.


 * $$ \int_0^{1} \frac{\cos{x}}{\sqrt{x}}\,dx $$



\begin{align} \int_0^{1} \frac{\cos{x}}{\sqrt{x}}\,dx & {} = \int_0^{1} \frac{\cos{u^2}}{u}(2u)\,du \\ & {} = \int_0^{1} \cos{u^2}\,du \end{align} $$