1.

`int_(0)^(pi//4) (sec^(2)x)/((1+tan x)(2+tan x))dx` is equal toA. `log_(e)((2)/(3))`B. `log_(e)3`C. `(1)/(3)log_(e)((4)/(3))`D. `log_(e)((4)/(3))`

Answer» Correct Answer - D
Put `1+tanx=t rArr sec^(2)xdx=dt`
`therefore" "int_(0)^(pi//4)(sec^(2)x)/((1+tanx)(2+tanx))dx`
`=int_(1)^(2)(dt)/(t(1+t))=int_(1)^(2)(dt)/(t)-int_(1)^(2)(dt)/(1+t)`
`=[logt-log(1+t)]_(1)^(2)`
`=log_(2)2-log_(e)3+log_(e)2=log_(e).(4)/(3)`.


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