1.

` int _(0)^(pi//4) log ((sin x+ cos x)/(cosx))dx ` is equal toA. `(pi)/(8)log2`B. `(pi)/(4)log2`C. `log2`D. `(pi)/(2)log2`

Answer» Correct Answer - A
Let ` l = int _(0)^(pi//4) log ((sin x + cos x)/(cos x)) dx`
` l =- int _(0)^(pi//4) log (1+tan x ) dx" " `
` = int _(0)^(pi//4) log [ 1 + tan (pi/4 -x) ] dx`
` = int _(0)^(pi//4) log (1+ (tan pi//4-tan x)/(1+tan pi //4 tanx))dx`
` = int _(0)^(pi//4) log (1+ (1-tanx)/(1+tan x)) dx`
` rArr l = int _(0)^(pi//4) log (2/(1+tan x))dx" "` ...(ii)
On adding Eqs. (i) and (ii) , we get
` 2l = int _(0)^(pi//4) log 2 dx = log 2 (pi/4)`
` rArr l = pi/8 log 2 `


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