2022 HCI Promo Q8
(a)
Find $\int{3t{{\tan }^{-1}}\left( 3t \right)\text{d}t}$.
[4]
(b)
Using the substitution $u={{x}^{2}}+1$, show that $\int_{0}^{\sqrt{7}}{{{x}^{3}}{{\left( {{x}^{2}}+1 \right)}^{\frac{1}{3}}}\text{d}x}$ can be expressed as $\frac{1}{2}\int_{a}^{b}{{{u}^{\frac{4}{3}}}-{{u}^{\frac{1}{3}}}}\text{ d}u$, where $a$ and $b$ are constants to be determined.
Hence find the exact value of $\int_{0}^{\sqrt{7}}{{{x}^{3}}}{{\left( {{x}^{2}}+1 \right)}^{\frac{1}{3}}}\text{d}x$.
[5]
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