∫ dx/[ (sinx)^4 + (cosx)^4 ]

分子分母同時除以 (cosx)^4

=∫ (secx)^4/[ 1+(tanx)^4 ] dx

=∫ (secx)^2/[ 1+(tanx)^4 ] dtanx

=∫ [ 1+ (tanx)^2] /[ 1+(tanx)^4 ] dtanx

u=tanx

=∫ ( 1+ u^2) /( 1+u^4 ) du

分子分母同時除以 u^2

=∫ ( 1+ 1/u^2) /(u^2+ 1/u^2 ) du

=∫ ( 1+ 1/u^2) /[ (u -1/u)^2 +2 ] du

d( u- 1/u) = ( 1+ 1/u^2) du

=∫ d( u- 1/u) /[ (u -1/u)^2 +2 ]

=(1/√2 )∫ d[( u- 1/u)/√2] /{ 1 + [(u -1/u)/√2]^2 }

=(1/√2 )arctan[( u-1/u)/√2] + C

=(1/√2 )arctan[( tanx -1/tanx)/√2] + C

∫(sinx)^4dx =∫[(1/2)(1-cos2x]^2dx =(1/4)∫[1-2cos2x+(cos2x)^2]dx =(1/4)∫[1-2cos2x+(1/2)(1+cos4x)]dx =(3/8)∫dx-(1/2)∫cos2xdx+(1/8)∫cos4xdx =(3/8)∫dx-(1/4)∫cos2xd2x+(1/32)∫cos4xd4x =(3/8)x-(1/4)sin2x+(1/32)sin4x+C