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Evaluate the following : int(0)^(pi//2...

Evaluate the following : `int_(0)^(pi//2)(sinx-cosx)/(1+sinx.cosx)dx`

Text Solution

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Let `I=int_(0)^(pi//2)(sinx-cosx)/(1+sinx.cosx)dx`
We use the property, `int_(0)^(a)f(x)dx=int_(0)^(a)f(a-x)dx.`
Here `a=(pi)/(2).` Hence changing x by `(pi)/(2)-x,` we get,
`I=int_(0)^(pi//2)(sin((pi)/(2)-x)-cos((pi)/(2)-x))/(1+sin((pi)/(2)-x).cos((pi)/(2)-x))dx`
`=int_(0)^(pi//2)(cosx-sinx)/(1+cosx.sinx)dx`
`=-int_(0)^(pi//2)(sinx-cosx)/(1+sinx.cosx)dx=-I`
`therefore" "2I=0" "therefore I=0.`
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