Suppose we define the definite integral using the following formula ${\int }_a^{b}f\left(x\right)dx=\frac{b-c}{2}\left(f\left(a\right)+f\left(b\right)\right)$ , for more accurate result for $c\in \left(a,b\right)F\left(c\right)=\frac{b-c}{2}\left(f\left(a\right)-f\left(c\right)\right)+\frac{b-c}{2}\left(f\left(b\right)-f\left(c\right)\right)$. When $c=\frac{a+b}{2},{\int }_a^{b}f\left(x\right)dx=\frac{b-a}{4}(f\left(a\right)+f\left(b\right)+f\left(2f\left(c\right)\right)$ then ${\int }_0^{\frac{\pi }{2}}\sin xdx$

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