Question

∫sinx ∙ cosx ∙ cos2x ∙ cos4x ∙ cos8x ∙ cos16xdx = _____ + c

(a) [(sin16x)/(1024)]

(b) – [(cos32x)/(1024)]

(c) [(cos32x)/(1096)]

(d) – [(cos32x)/(1096)]

Answer 1

The correct option (b) – [(cos32x)/(1024)]  

Explanation:

We know sin16x = sin[2(8x)]

= 2 sin8x ∙ cos8x 

= 2 sin[2(4x)] cos8x 

= 2(2)sin(4x) ∙ cos4x ∙ cos8x 

= 4 × 2 sin2x ∙ cos2x ∙ cos4x ∙ cos8x 

= 8 × 2 × sinx cosx ∙ cos2x ∙ cos4x ∙ cos8x sin16x 

= 16sinx ∙ cosx ∙ cos2x ∙ cos4x ∙ cos8x 

∴ cosx ∙ cos2x ∙ cos4x ∙ cos8x 

= [(sin16x)/(16sinx)] (1) 

I = ∫sinx ∙ [(sin16x)/(16sinx)] ∙ cos16x ∙ dx 

= (1/16) × (1/2) ∫sin32x dx

= [(– cos32x)/(32)²]

= [(– cos32x)/(1024)]