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The area of the region bounded by the cu...

The area of the region bounded by the curves `y=sqrt[[1+sinx]/cosx]` and `y=sqrt[[1-sinx]/cosx]` bounded by the lines x=0 and `x=pi/4` is

Text Solution

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`sinx=(2tanx/2)/(1+tan^2x/2)`
`cosx=(1-tan^2x/2)/(1+tan^2x/2)`
`y=sqrt((1+((2tanx/2)/(1+tan^2x/2))/((1-tan^2x/2)/(1+tan^2x/2)))`
`y=(1+tanx/2)/(sqrt(1-tan^2x/2)`
`y=(1-tanx/2)/sqrt(1-tan^2x)`
`I=int_0^(pi/4)sqrt((1+sinx)/(cosx))-sqrt((1-sinx)/(cosx))dx`
`I=int_0^(pi/4)(2tan(x/2))/sqrt(1-tan^2x/2)dx`
`I=int_0^(sqrt2-1)(2t2dt)/(sqrt(1-t^2)(1+t^2))`
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