Question

∫[(cos⁸x – sin⁸x)/(1 – 2sin²xcos²x)]dx = ______ + c

(a) – [(cos2x)/2] 

(b) – [(sin2x)/2]

(c) [(cos2x)/2]

(d) [(sin2x)/2]

Answer 1

The correct option (d) [(sin2x)/2]   

Explanation:

I = [(cos⁸x – sin⁸x)/(1 – 2sin²xcos²x)]dx  ...(1) 

consider cos⁸x – sin⁸x = (cos⁴x – sin⁴x) ∙ (cos⁴x + sin⁴x) 

= (cos²x – sin²x)(cos²x + sin²x)(1 – 2sin²x ∙ cos²x) 

= (cos²x – sin²x)(1 – 2sin²x ∙ cos²x) 

∴ putting this value in (1), 

I = ∫[{(cos²x – sin²x)(1 – 2sin²x ∙ cos²x)}/(1 – 2sin²x ∙ cos²x)] ∙ dx 

= ∫(cos²x – sin²x) 

= ∫cos2xdx 

I = [(sin2x)/2] + c