Application Of Integration

Suppose we define the definite integral using the following formula \(\int_\limits{a}^{b} f(x) d x=\frac{b-a}{2}(f(a)+f(b))\), for more accurate result for

\(c \in(a, b) \mathrm{F}(c)=\frac{c-a}{2}(f(a)+f(c))+\frac{b-c}{2}(f(b)+f(c))\).

When \(c=\frac{a+b}{c}, \int_\limits{a}^{b} f(x) d x=\frac{b-a}{4}(f(a)+f(b)+2 f(c))\)

\(\int_\limits{0}^{\pi / 2} \sin x d x\) is equal to: