一道不定积分∫cos∧4xsin∧2xdx

Question

一道不定积分
∫cos∧4xsin∧2xdx

Answer 1

∵cos^4xsin^2x
  =cos^4x(1-cos²x)
  =cos^4x-cos^6x
  =[1+cos(2x)]²/4-[1+cos(2x)]³/8
  =[1+2cos(2x)+cos²(2x)]/4-[1+3cos(2x)+3cos²(2x)+cos³(2x)]/8
  =1/8+1/8cos(2x)-1/8cos²(2x)-1/8cos³(2x)
  =1/8+1/8cos(2x)-[1+cos(4x)]/16-1/8cos³(2x)
  =1/16+1/8cos(2x)-cos(4x)/16-1/8cos³(2x)
∴∫cos^4xsin^2xdx
  =∫[1/16+1/8cos(2x)-cos(4x)/16-1/8cos³(2x)]dx
  =x/16+sin(2x)/16-sin(4x)/64-1/16∫[1-sin²(2x)]d[sin(2x)]
  =x/16+sin(2x)/16-sin(4x)/64-[sin(2x)-sin³(2x)/3]/16+C
  =x/16-sin(4x)/64+sin³(2x)/48+C