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Evaluate: ∫ (((cos x – sin x)/ (cos x + sinx)) × (2 + 2sin 2 x)) dx

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∫ ((cos x – sin x)/ (cos x + sin x)× (2 + 2sin 2x) dx 

= ∫ (((cos x – sinx)/ (cos x + sinx )) (1 + (sinx + cos x)2)) dx 

= ∫ (1 + t2/ t) dt, Let cos x + sin x = t 

= (– sin x + cos x) dx = dt 

= ∫ (t + (1/t)) dt 

= (t2/2) + log |t| + c 

= ((sin x + cos x)2/ 2) + log |sin x + cos x| + c

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