1.

Which of the following option(s) is/are incorrect?

Answer»

`int_(0)^(pi//8)|(SIN(8nx)+cos(8nx))/x| dx lt(2sqrt(2))/(pi)(1+1/2+1/3+………+1/n)`
`2/(IN3) lt int_(0)^(1)3^(x^(1//3)) dx lt int_(0)^(1) 4^(x^(1//3))dx`
`int_(0)^(pi//6) sqrt(sinx)(8-3sqrt(sinx))dx GT (8pi)/9`
`1/4 lt int_(pi//4)^(pi//3)(tanx)/x dx lt 1/(sqrt(3))`

Solution :(A) Let `int_(0)^(npi)|(sint+cost)/t|dt=A_(n)`
`A_(n)gt 1/(pi) [int_(0)^(pi)"sinnt+cost|dt+int_(pi)^(2pi)(|sinnt+cost|dt)/2+..]`
`gt1/(pi)(1+1/2+1/3+…..+1/n)(int_(0)^(pi)|sint+cost|dt)`
`gt(2sqrt(2))/(pi)(1+1/2+1/3+….31/n)`
(C) `(3sqrt(sinx)+8-3sqrt(sinx))/2gesqrt(3sqrt(sinx)(8-3sqrt(sinx)))`
(D) `1le tan x le sqrt(3)` and `3/(pi) le 1/x le 4/(pi)`
So, `3/(pi)le (tanx)/x le (4sqrt(3))/(pi)`


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