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Chain Rule & Integration by Substitution

Review question

Can we find $\int 1/(\sqrt{x+2} - \sqrt{x+1}) dx$?
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Ref: R9701

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Question

By using the substitution \(x = \cos \theta\), or otherwise, evaluate \[\begin{equation*} \int_0^1 \frac{1}{(1-x)^{1/2}(1+x)^{3/2}} \:dx. \end{equation*}\]
Find \[\begin{align*} \text{(i)}\; &\int \frac{1}{\sqrt{x+2} - \sqrt{x+1}} \:dx, \\ \text{(ii)}\; &\int \frac{x}{(x+1)(x+2)^2} \:dx. \end{align*}\]