10 Let \(I_{n}=\int_{\frac{1}{4} \pi}^{\frac{1}{2} \pi} \cot ^{n} x \mathrm{~d} x\), where \(n \geqslant 0\).
(i) By considering \(\frac{\mathrm{d}}{\mathrm{d} x}\left(\cot ^{n+1} x\right)\), or otherwise, show that
\(I_{n+2}=\frac{1}{n+1}-I_{n} .\)

The curve \(C\) has equation \(y=\cot x\), for \(\frac{1}{4} \pi \leqslant x \leqslant \frac{1}{2} \pi\).
(ii) Find, in an exact form, the \(y\)-coordinate of the centroid of the region enclosed by \(C\), the line \(x=\frac{1}{4} \pi\) and the \(x\)-axis.

The curve \(C\) has equation \(y=x^{a}\) for \(0 \leqslant x \leqslant 1\), where \(a\) is a positive constant. Find, in terms of \(a\), the coordinates of the centroid of the region enclosed by \(C\), the line \(x=1\) and the \(x\)-axis.

(i) Using the substitution \(u=\tan x\), or otherwise, find \(\int \sec ^{2} x \tan ^{2} x \mathrm{~d} x\).

It is given that, for \(n \geqslant 0\),
\(I_{n}=\int_{0}^{\frac{1}{4} \pi} \sec ^{n} x \tan ^{2} x \mathrm{~d} x\)
(ii) Using the result that \(\frac{\mathrm{d}}{\mathrm{d} x}(\sec x)=\tan x \sec x\), show that, for \(n \geqslant 2\),
\((n+1) I_{n}=(\sqrt{ } 2)^{n-2}+(n-2) I_{n-2} .\)

(iii) Hence find the mean value of \(\sec ^{4} x \tan ^{2} x\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant \frac{1}{4} \pi\), giving your answer in exact form.

Find the exact value of \(\int_{2}^{\frac{7}{2}} \frac{1}{\sqrt{4 x-x^{2}-1}} \mathrm{~d} x\).

Find the value of \(\int_{6}^{7} \frac{1}{\sqrt{(x-5)^{2}-1}} \mathrm{~d} x\), giving your answer in the form \(\ln (a+\sqrt{b})\), where \(a\) and \(b\) are integers to be determined.

The curve \(C\) has equation \(y=\mathrm{e}^{-2 x}\). Find, giving your answers correct to 3 significant figures,
(i) the mean value of \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) over the interval \(0 \leqslant x \leqslant 2\),

(ii) the coordinates of the centroid of the region bounded by \(C, x=0, x=2\) and \(y=0\).

Let \(y=\mathrm{e}^{x}\). Find the mean value of \(y\) with respect to \(x\) over the interval \(0 \leqslant x \leqslant 2\).

Show that the mean value of \(x\) with respect to \(y\) over the interval \(1 \leqslant y \leqslant \mathrm{e}^{2}\) is \(\frac{\mathrm{e}^{2}+1}{\mathrm{e}^{2}-1}\).

The equation of a curve is \(y=\lambda x^{2}\), where \(\lambda\gt 0\). The region bounded by the curve, the \(x\)-axis and the line \(x=a\), where \(a\gt 0\), is denoted by \(R\). The \(y\)-coordinate of the centroid of \(R\) is \(a\). Show that \(\lambda=\frac{10}{3 a}\).

Let
\(I_{n}=\int_{0}^{\frac{1}{2} \pi} \sin ^{n} \theta \mathrm{~d} \theta\)
where \(n\) is a non-negative integer. Show that \(I_{n+2}=\frac{n+1}{n+2} I_{n}\).

The region \(R\) of the \(x-y\) plane is bounded by the \(x\)-axis, the line \(x=\frac{\pi}{2 m}\) and the curve whose equation is \(y=\sin ^{4} m x\), where \(m\gt 0\). Find the \(y\)-coordinate of the centroid of \(R\).