2022 NJC Promo Q6
(i)
Use the substitution $x=m\tan t$ to find $\int{\frac{1}{\sqrt{{{m}^{2}}+{{x}^{2}}}}\text{d}x}$, where $m$ is a positive constant and $0\le t<\frac{\pi }{2}$.
[4]
(ii)
Find $\int{\frac{x}{\sqrt{{{m}^{2}}+{{x}^{2}}}}\text{d}x}$.
[2]
(iii)
Hence, find $\int{\frac{x}{\sqrt{{{m}^{2}}+{{x}^{2}}}}{{\tan }^{-1}}\left( \frac{x}{m} \right)}\,\text{d}x$.
[3]
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