<ol class="abc-list"><li><div class="list-item-paragraph">Find $\int{3t{{\tan }^{-1}}\left( 3t \right)\mathrm{d}t}$.</div><p style="text-align: right;">[4]</p></li><li><div class="list-item-paragraph">Using the substitution $u={{x}^{2}}+1$, show that $\int_{0}^{\sqrt{7}}{{{x}^{3}}{{\left( {{x}^{2}}+1 \right)}^{\frac{1}{3}}}\mathrm{d}x}$ can be expressed as $\frac{1}{2}\int_{a}^{b}{{{u}^{\frac{4}{3}}}-{{u}^{\frac{1}{3}}}}\mathrm{ d}u$, where $a$ and $b$ are constants to be determined.</div><div class="list-item-paragraph">Hence find the exact value of $\int_{0}^{\sqrt{7}}{{{x}^{3}}}{{\left( {{x}^{2}}+1 \right)}^{\frac{1}{3}}}\mathrm{d}x$.</div><p style="text-align: right;">[5]</p></li></ol>

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

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

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

2022 HCI Promo Q8

2022 HCI Promo Q8

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