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Question
By using the properties of the definite integral, evaluate the integral:
`int_0^(pi/2) (sin x - cos x)/(1+sinx cos x) dx`
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Solution
`I = int_0^(pi/2) (sin x - cosx)/(1+sinx cos x) dx` ....(i)
`I = int_0^(pi/2) (sin (pi/2-x)-cos(pi/2-x))/(1 + sin(pi/2-x)cos(pi/2-x))dx`
`I = int_0^(pi/2) (cosx-sinx)/(1+cosxsinx)dx` .....(ii)
Adding (i) and (ii), we get :
`2 I = int_0^(pi/2) ((sin x - cos x)/ (1 + sin x cos x) + (cos x - sin x)/ (1 + sin x cos x)) dx`
`2I = int_0^(pi/2)(sinx-cosx+ cosx - sinx)/(1 +sinxcosx) dx`
`2I = 0 ⇒I=0`
`⇒ int_0^(pi/2) (sinx-cosx)/(1+sinxcosx) dx=0`
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