2022 HCI Promo Q8

Find \(\displaystyle\displaystyle \displaystyle\int{3t{{\tan }^{-1}}\left( 3t \right)\mathrm{d}t}\).[4]

Using the substitution \(u={{x}^{2}}+1\), show that \(\displaystyle\displaystyle \displaystyle\int_{0}^{\sqrt{7}}{{{x}^{3}}{{\left( {{x}^{2}}+1 \right)}^{\frac{1}{3}}}\mathrm{d}x}\) can be expressed as \(\frac{1}{2}\displaystyle\displaystyle \displaystyle\int_{a}^{b}{{{u}^{\frac{4}{3}}}-{{u}^{\frac{1}{3}}}}\mathrm{ d}u\), where \(a\) and \(b\) are constants to be determined.

Hence find the exact value of \(\displaystyle\displaystyle \displaystyle\int_{0}^{\sqrt{7}}{{{x}^{3}}}{{\left( {{x}^{2}}+1 \right)}^{\frac{1}{3}}}\mathrm{d}x\).[5]