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I =int (sin ^(8) x- cos ^(8) x)/( 1-2 si...

`I =int (sin ^(8) x- cos ^(8) x)/( 1-2 sij ^(2) x cos ^(2)x ) dx` is equal to:

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`int (sin^(8) x-cos^(8) x)/(1-2sin^(2) cos^(2)x) dx`
`=int ({(sin^(4)x)^(2)-(cos^(4)x)^(2)})/(1-2sin^(2) xcos^(2)x)dx`
`=int ((sin^(4) x-cos^(4)x)(sin^(4)x+cos^(4)x))/(1-2sin^(2)xcos^(2)x)dx`
`(sin^(2)x-cos^(2)x)(sin^(2)x+cos^(2)x)`
`=int ("["(sin^(2) x)^(2)+(cos^(2)x)^(2)"]")/(1-2sin^(2)x cos^(2)x)dx`
`(sin^(2)x-cos^(2)x)1"{"(sin^(2)x+cos^(2)x)^(2)`
`=int (-2sin^(2)x.cos^(2)x)/(1-2sin^(2)xcos^(2)x)dx`
`=int (-cos 2x(1-2sin^(2))x.cos^(2)x)/(1-2sin^(2)xcos^(2)x)dx`
`=int (-cos 2x(1-2sin^(2)x.cos^(2)x))/(1-2sin^(2)x cos^(2)x)dx`
`=-int cos 2x dx= -.(sin 2x)/(2)+C`
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