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Prove that (1+sqrt(2))/2lt int_(0)^(pi//2)(sinx)/x dx lt (pi+2sqrt(2))/4 |
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Answer» SOLUTION :`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` |
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