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Question
Integrate the function:
`(sin^8 x - cos^8 x)/(1-2sin^2 x cos^2 x)`
Solution
Let `I = int (sin^8 x - cos^8 x)/ (1 - 2 sin^2x cos^2 x) dx`
We have, (sin8x - cos8x)
= (sin4x + cos4x) (sin4x - cos4x)
= [(sin2x + cos2x)2 - 2 sin2x cos2 x] (sin2 x + cos2 x) (sin2 x - cos2 x)
= (1 - 2 sin2x cos2x) (1) (-cos2x)
∴ `I = int ((1 - 2 sin^2 x cos^2 x) (- cos 2x))/(1 - 2 sin^2 x cos x) dx`
`= - int cos 2x dx`
`= -1/2 sin2x + C`
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