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int0^a log (cota+ tanx)dx where a in (0,...

`int_0^a log (cota+ tanx)dx` where `a in (0,pi/2)` is

A

(a) `a ln (sina)`

B

(b) `-a ln (sina)`

C

(c) `-a ln (cos a)`

D

(d) none of these

Text Solution

Verified by Experts

The correct Answer is:
B
`I=int_(0)^(a)ln(cot a +tanx)dx`
`=int_(0)^(a)ln((cos(a0x))/(sina cosx))dx" (1)"`
`therefore" "I=int_(0)^(a)ln((cosx)/(sina cos(a-x)))dx" (2)"` Adding (1) and (2) we get `2I=int_(0)^(a)ln((1)/(sin^(2)a))dx`
`=-2int_(1)^(a)ln(sina)dx`
`=-2aln(sina)`
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