2018 DHS J2 MYCT P1 Q5

Show that \(\frac{\mathrm{d}}{\mathrm{d}x}\left[ {{\left( \sqrt{x} \right)}^{x}} \right]=\frac{1}{2}{{\left( \sqrt{x} \right)}^{x}}\left( 1+\ln x \right)\).[4]

Hence by using the substitution \(u=\sqrt{x}\), evaluate \(\displaystyle\displaystyle \displaystyle\int_{1}^{2}{{{\left( u \right)}^{{{u}^{2}}+1}}}\left( 1+2\ln u \right)\ \mathrm{d}u\) exactly.[4]