令x=tanz,dx=sec²zdz
∫(0→1)xarctanx/(1+x²)³dx
=∫(0→π/4)ztanz/sec⁶z*(sec²zdz)
=∫(0→π/4)zsinzcos³zdz
=∫(0→π/4)zcos³zd(-cosz)
=(-1/4)∫(0→π/4)zd(cos⁴z)
=(-1/4)zcos⁴z|(0→π/4)+(1/4)∫(0→π/4)cos⁴zdz
=-π/64+(1/4)∫(0→π/4)[(1+cos2z)/2]²dz
=-π/64+(1/4)²∫(0→π/4)[1+2cos2z+(1+cos4z)/2]dz
=-π/64+(1/16)[z+sin2z+z/2+(1/8)sin4z]|(0→π/4)
=-π/64+(1/128)(8+3π)
=(8+π)/128