∫x²tan²xdx=∫x²(sec²-1)dx=∫x²sec²dx-∫x²dx
=∫x²dtanx-x³/3+C
=x²tanx+∫tanxdx²-x²/3+C
=2∫xtanxdx+x²tanx-x²/3+C
∫x*tanxdx=∫xsinx/cosx dx=-∫xd(cosx)/cosx=-∫(π/2-(π/2-x))d(sin(π/2-x))/sin(π/2-x)
设t=sin(π/2-x)
原式=-∫(π/2-arccost)/t dt=-=∫π/2t dt+∫arccost/t dt=-π/2*lnt+∫arccost/t dt
根据泰勒级数
arccost=x+x^3/(2*3)+1*3*x^5/(2*4*5).
所以原式=-π/2*lnt+∫arccost/t dt=π/2*lnt+∫1+x^2/(2*3)+1*3*x^4/(2*4*5).dt=
-π/2*lnt+x+x^3/(2*3*3)+1*3*x^5/(2*4*5*5).