x=tanu
dx=(secu)^2 du
∫(0->1) (1+x)^3/(1+x^2)^2 dx
=∫(0->π/4) (1+tanu)^3/(secu)^2 du
=∫(0->π/4) [ 1+3tanu + 3(tanu)^2 +(tanu)^3]/(secu)^2 du
=∫(0->π/4) [ (cosu)^2 +3sinu.cosu + 3(sinu)^2 + (sinu)^3/cosu ] du
=∫(0->π/4) [ 1 +3sinu.cosu + 2(sinu)^2 + (sinu)^3/cosu ] du
=∫(0->π/4) [ 1 +(3/2)sin2u + (1-cos2u) ] du +∫(0->π/4) (sinu)^3/cosu du
=∫(0->π/4) [ 2 +(3/2)sin2u -cos2u ] du -∫(0->π/4) (sinu)^2/cosu dcosu
=[2u-(3/4)cos2u -(1/2)sin2u]|(0->π/4) +∫(0->π/4) [(cosu)^2 -1]/cosu dcosu
=[ (π/2-0-1/2)-(0-3/4-0)] + [ (1/2)(cosu)^2 - ln|cosu|]|(0->π/4)
=π/2 +1/4 + [ ( 1/4 +(1/2)ln2) - (1/2-0) ]
=π/2 +(1/2)ln2
dx=(secu)^2 du
∫(0->1) (1+x)^3/(1+x^2)^2 dx
=∫(0->π/4) (1+tanu)^3/(secu)^2 du
=∫(0->π/4) [ 1+3tanu + 3(tanu)^2 +(tanu)^3]/(secu)^2 du
=∫(0->π/4) [ (cosu)^2 +3sinu.cosu + 3(sinu)^2 + (sinu)^3/cosu ] du
=∫(0->π/4) [ 1 +3sinu.cosu + 2(sinu)^2 + (sinu)^3/cosu ] du
=∫(0->π/4) [ 1 +(3/2)sin2u + (1-cos2u) ] du +∫(0->π/4) (sinu)^3/cosu du
=∫(0->π/4) [ 2 +(3/2)sin2u -cos2u ] du -∫(0->π/4) (sinu)^2/cosu dcosu
=[2u-(3/4)cos2u -(1/2)sin2u]|(0->π/4) +∫(0->π/4) [(cosu)^2 -1]/cosu dcosu
=[ (π/2-0-1/2)-(0-3/4-0)] + [ (1/2)(cosu)^2 - ln|cosu|]|(0->π/4)
=π/2 +1/4 + [ ( 1/4 +(1/2)ln2) - (1/2-0) ]
=π/2 +(1/2)ln2
