\(\int \frac{x^3}{x^4 + 3x^2 + 2}dx\)
\(x^2 =u\)
\(2xdx = du\)
\(\int \frac{x^2 . x\,dx}{(x^2)^2 + 3x^2 + 2}\)
\(=\frac 12 \int \frac {u\, du}{u^2 + 3u + 2} \)
\(= \frac 12 \int \frac u{(u + 1)(u + 2)} du\)
\(=\frac12 \int \left( \frac 2{u + 2} - \frac 1{u + 1}\right)du\)
\(= \ln |u + 2| - \frac12 \ln |u + 1| + C\)
\(= \ln |x^2 + 2| - \frac 12 \ln|x^2 + 1|+C\)
