Prove that `(1+sqrt(2))/2lt int_(0)^(pi//2)(sinx)/x dx lt (pi+2sqrt(2))/4`

Explanation

`f(x)=(sin x)/x`
`:. f'(x)=(xcosx-sinx)/(x^(2))=(cosx(x-tanx))/(x^(2))`
For `xepsilon(0,pi//2),xlt tanx`
`,.f'(x)lt0`
So, `f(x)` is decreasing function.
`lim_(xto0)(sinx)/x=1` and `lim_(to pi//2)(sinx)/x=2/(pi)`
So the graph of the function is as shown in the following figure:

From the figure,
Area of rectangle `OJED+` Area of rectangle `JAGH`
`ltint_(0)^(pi//2)(sinx)/x dx`
`lt` Area of `OJKC+` Area of `JAFE`
`:.(pi)/4 . 4/(pisqrt(2))+(pi)/4 . 2/(pi) lt int_(0)^(pi//2) (sinx)/x dx lt (pi)/4 . 1+(pi)/4 . 4/(pisqrt(2))`
`implies 1/(sqrt(2))+1/2 lt int_(0)^(pi//2)(sinx)/x dx lt (pi)/4 + 1/(sqrt(2))`
`implies(1+sqrt(2))/2 lt int_(a)^(pi//2)(sinx)/x dx lt(pi+2sqrt(2))/4`