\(\int \frac{x^3}{(x^2 + 1)^3 }dx = \int \frac{x^2.xdx}{(x^2 + 1)^3}\)
\(1 + x^2 = t\)
\(2xdx = dt\)
\(xdx = \frac{dr}2\)
\(\therefore \int \frac{x^3}{(x^2 + 1)^3}dx = \frac12 \int\frac{(t - 1)dt}{t^3}\)
\( = \frac12 \int (t^{-2} - t^{-3} )dt\)
\( =\frac12 \left(- \frac1t\ - \frac1{2t^2}\right)+ C\)
\(= - \frac14 \left(\frac{2t + 1}{t^2}\right)+ C\)
\(= - \frac14\left(\frac{2(1 + x^2)+1}{(1 +x^2)^2}\right)+ C\) \((\because t = 1 + x^2)\)
\(= -\frac{2x^2 + 3}{4(1 + x^2)^2} + C\).
