Question

Suppose we define definite integral using the formula ${\int }_a^{b}f\left(x\right)dx=\frac{b-a}{2}\{f\left(a\right)+f\left(b\right)\}.$ For more accurate result, we have ${\int }_a^{b}f\left(x\right)dx=\frac{b-a}{4}\{f\left(a\right)+f\left(b\right)+2f\left(c\right)\}$, when $c=\frac{a+b}{2}.$ Also, let $F\left(c\right)=\frac{c-a}{2}\{f\left(a\right)+f\left(c\right)\}+\frac{b-c}{2}\{f\left(b\right)+f\left(c\right)\}$, when $cϵ(a,b).$

(i) ${\int }_0^{\pi /2}\sin xdx$ equals

• A. $\frac{\pi }{8}(1+\sqrt{2})$
• B. $\frac{\pi }{4}(1+\sqrt{2})$
• C. $\frac{\pi }{8\sqrt{2}}$
• D. $\frac{\pi }{4\sqrt{2}}$

Answer 1

The correct option is A $\frac{\pi }{8}(1+\sqrt{2})$

${\int }_0^{\frac{\pi }{2}}\sin xdx=\frac{\frac{\pi }{2}-0}{4}(\sin 0+\sin \left(\frac{\pi }{2}\right)+2\sin \left(\frac{0+\frac{\pi }{2}}{2}\right))$
$=\frac{\pi }{8}(1+\sqrt{2})$