①
e^(\(x^{2}\))=t とおくと
\(x^{2}\)=logt
(2x)e^(\(x^{2}\))dx=dt
(\(x^{3}\))e^(\(x^{2}\))dx
=(\(x^{2}\))(\(\frac{1}{2}\))(2x)e^(\(x^{2}\))dx
=(\(\frac{1}{2}\))logt dt
よって
与式=(\(\frac{1}{2}\))∫logt dt
=(\(\frac{1}{2}\))(tlogt-t)+C
=(\(\frac{1}{2}\))e^(\(x^{2}\))(\(x^{2}\)-1)+C・・・（答）

②
(\(x^{2}\)-x+1) /(x-1)(x-2)(x-3)
={A/(x-1)}+{B/(x-2)}{C/(x-3)}とおく
A(x-2)(x-3)+B(x-1)(x-3)+C(x-1)(x-2)=\(x^{2}\)-x+1となります。
あとは展開・整理・係数比較して
A+ B+ C=1
5A+4B+3C=1
6A+3B+2C=1を解いて　A=\(\frac{1}{2}\) B=-3 C=\(\frac{7}{2}\)
よって与式は
∫[1/{2(x-1)}-3/(x-2)+7/{2(x-3)}]dx
=(\(\frac{1}{2}\))log|x-1|-3log|x-2|+(\(\frac{7}{2}\))log|x-3|+C・・（答）