[arctan(x²/2)]'
=1/[1+(x²/2)²]•(x²/2)'
=1/(1+x⁴/4)•(2x/2)
=4/(4+x⁴)•x
=4x/(4+x⁴)
∫x/(4+x²)dx=(1/2)∫1/(4+x²)d(x²+4)=(1/2)ln(x²+4)+C
∫x/(4+x⁴)dx=(1/2)∫1/(4+x⁴)d(x²)=(1/2)∫1/[4+(x²)²]d(x²)=(1/2)(1/2)arctan(x²/2)=(1/4)arctan(x²/2)+C或用第一个结果:[arctan(x²/2)]'=4x/(4+x⁴)==>arctan(x²/2)=∫4x/(4+x⁴)dx==>(1/4)arctan(x²/2)=∫x/(4+x⁴)dx==>∫x/(4+x⁴)dx=(1/4)arctan(x²/2)+C