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`int[sqrt(cotx)+sqrt(tan x)]dx` ज्ञात कीजिए!A. `sqrt2tan^(-1)((tanx)/(sqrt(2 tan x)))+C`B. `sqrt2tan^(-1)((tanx-1)/(sqrt(2 tan x)))+C`C. `(tanx)/(sqrt2)*tan^(-1)((cot x+1)/(sqrt(2 tanx)))+C`D. `(tanx)/(sqrt2)*tan^(-1)((cotx+1)/(sqrt(tanx)))+C`

Answer» Correct Answer - B
Let `I=int((sin x+cosx))/(sqrtsinx*cos x)dx`
`=int(sqrt2(sinx+cos x))/(sqrt(2sin x*cos x))dx`
`=sqrt2 int(sin x+cos x)/(sqrtsin 2x)dx`
Put `sinx -cos x=t`
`implies(cos x+sin x)dx=dt`
Also, `sin 2x=(1-t^(2))`
`thereforeI=sqrt2int(dt)/(sqrt(1-t^(2)))`
`=sqrt2sin ^(-1)t+c`
`=sqrt2sin ^(-1)(sinx-cos x)+c`
`=sqrt2 tan ^(-1)((tan x-1)/(sqrt(2 tan x)))+c`


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