Question

integrate (x^4 tanx)/(1+x^2)

Answer 1

use partial integration\[x^4tanx/1+x^2=ax+b/1+x^2\]let x=0, then b=0
let x=pi/4 then a=(pi/4)^3
\[\int\limits(ax+b)dx/(1+x^2)\]let x=tan(u)
dx=sec^2(u)du
remember that tan^2 +1 =sec^2\[\int\limits((\pi/3)^3tanu\sec^2udu/\sec^2u\]\[(\pi/3)^3\int\tan(u)du=(\pi/3)^3ln(cos(u))\]\[(\pi/3)^3ln(cos(\tan^{-1} x))\]ans=\[(\pi/3)^3ln(cos(1/(1+x^2)))\]