Evaluate the following integral.

Question

Evaluate the following integral.

 

 

∫ sin^2 x cos^2 x dx

Answer 1

 

Hi Daniel,

 

∫ sin²(x)·cos²(x) dx = ∫ (sin(x)·cos(x))² dx

 

                             = ∫ ((1/2)sin(2x))² dx   (from the identity, sin(2x) = 2sin(x)·cos(x))

 

                             = (1/4) ∫ sin²(2x) dx

 

                             = (1/4) ∫ (1 - cos(4x))/2 dx   (from the identity, cos(4x) = 1 - 2sin²(2x))

 

                             = (1/8) ∫ (1 - cos(4x)) dx

 

                             = (1/8)·(x - (1/4)sin(4x)) + C

 

                             = x/8 - sin(4x)/32 + C   (C is an arbitrary constant)