The integral \(I_{n}\), where \(n\) is an integer, is defined by \(I_{n}=\int_{0}^{\frac{3}{2}}\left(4+x^{2}\right)^{-\frac{1}{2} n} \mathrm{~d} x\).
(a) Find the exact value of \(I_{1}\), expressing your answer in logarithmic form.

(b) By considering \(\frac{\mathrm{d}}{\mathrm{d} x}\left(x\left(4+x^{2}\right)^{-\frac{1}{2} n}\right)\), or otherwise, show that
\[4 n I_{n+2}=\frac{3}{2}\left(\frac{2}{5}\right)^{n}+(n-1) I_{n} .\]
(c) Find the value of \(I_{5}\).