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 \(R\).
\(4 \quad\) It is given that, for \(n \geqslant 0\),
\(I_{n}=\int_{0}^{1} x^{n} \mathrm{e}^{x^{3}} \mathrm{~d} x\)
(i) Show that \(I_{2}=\frac{1}{3}(\mathrm{e}-1)\).
(ii) Show that, for \(n \geqslant 3\),
\(3 I_{n}=\mathrm{e}-(n-2) I_{n-3} .\)
(iii) Hence find the exact value of \(I_{8}\).
The integral \(I_{n}\), where \(n\) is a positive integer, is defined by
\(I_{n}=\int_{\frac{1}{2}}^{1} x^{-n} \sin \pi x \mathrm{~d} x\)
(i) Show that
\(n(n+1) I_{n+2}=2^{n+1} n+\pi-\pi^{2} I_{n} .\)
(ii) Find \(I_{5}\) in terms of \(\pi\) and \(I_{1}\).
Answer only one of the following two alternatives.
EITHER
(i) Show that
\(\int_{-\frac12\pi}^{\frac12\pi} e^x\cos x\,dx=\frac12\left(e^{\frac12\pi}+e^{-\frac12\pi}\right).\)
(ii) It is given that, for \(n\ge 0\),
\(I_n=\int_{-\frac12\pi}^{\frac12\pi} e^{2x}\cos^n x\,dx.\)
Show that, for \(n\ge 2\),
\(4I_n=n(n-1)\int_{-\frac12\pi}^{\frac12\pi} e^{2x}\sin^2x\cos^{n-2}x\,dx-nI_n,\)
and deduce the reduction formula
\((n^2+4)I_n=n(n-1)I_{n-2}.\)
(iii) Using the result in part (i) and the reduction formula in part (ii), find the \(y\)-coordinate of the centroid of the region bounded by the \(x\)-axis and the arc of \(y=e^x\cos x\) from \(x=-\frac12\pi\) to \(x=\frac12\pi\). Give your answer correct to 3 significant figures.
OR
Let \(V\) be the subspace of \(\mathbb R^4\) spanned by
\(\mathbf v_1=\begin{pmatrix}1\\2\\0\\2\end{pmatrix},\quad \mathbf v_2=\begin{pmatrix}-2\\-5\\5\\6\end{pmatrix},\quad \mathbf v_3=\begin{pmatrix}0\\-3\\15\\18\end{pmatrix},\quad \mathbf v_4=\begin{pmatrix}0\\-2\\10\\8\end{pmatrix}.\)
(i) Show that the dimension of \(V\) is 3.
(ii) Express \(\mathbf v_4\) as a linear combination of \(\mathbf v_1\), \(\mathbf v_2\) and \(\mathbf v_3\).
(iii) Write down a basis for \(V\).
Let
\(\mathbf M=\begin{pmatrix}1&-2&0&0\\2&-5&-3&-2\\0&5&15&10\\2&6&18&8\end{pmatrix}.\)
(iv) Find the general solution of \(\mathbf M\mathbf x=\mathbf v_1+\mathbf v_2\).
The set of elements of \(\mathbb R^4\) which are not solutions of \(\mathbf M\mathbf x=\mathbf v_1+\mathbf v_2\) is denoted by \(W\).
(v) State, with a reason, whether \(W\) is a vector space.
Let \(I_{n}=\int_{0}^{1}(1-x)^{n} \sinh x \mathrm{~d} x\), where \(n\) is a non-negative integer.
(a) Show that, for \(n \geqslant 2, \quad I_{n}=-1+n(n-1) I_{n-2}\).
(b) Find the exact value of \(I_{2}\).
It is given that, for \(n \geqslant 0, I_{n}=\int_{0}^{\ln 3} \operatorname{sech}^{n} x \mathrm{~d} x\).
(a) Show that, for \(n \geqslant 2\),
\((n-1) I_{n}=\left(\frac{3}{5}\right)^{n-2}\left(\frac{4}{5}\right)+(n-2) I_{n-2} .\)
[You may use the result that \(\frac{\mathrm{d}}{\mathrm{d} x}(\operatorname{sech} x)=-\tanh x \operatorname{sech} x\).]
(b) Find the value of \(I_{4}\).
The integral \(I_{n}\), where \(n\) is an integer, is defined by \(I_{n}=\int_{0}^{\frac{4}{3}}\left(1+x^{2}\right)^{\frac{1}{2} n} \mathrm{~d} x\).
(a) Find the exact value of \(I_{-1}\) giving your answer in the form \(\ln a\), where \(a\) is an integer to be determined.
(b) By considering \(\frac{\mathrm{d}}{\mathrm{d} x}\left(x\left(1+x^{2}\right)^{\frac{1}{2} n}\right)\), or otherwise, show that
\((n+1) I_{n}=n I_{n-2}+\frac{4}{3}\left(\frac{5}{3}\right)^{n} .\)
(c) A curve has equation \(y=x^{2}\), for \(0 \leqslant x \leqslant \frac{2}{3}\). The arc length of the curve is denoted by \(s\).
Use the substitution \(u=2 x\) to show that \(s=\frac{1}{2} I_{1}\) and find the exact value of \(s\).
The integral \(I_{n}\) is defined by \(I_{n}=\int_{0}^{1}\left(1+x^{5}\right)^{n} \mathrm{~d} x\).
(a) By considering \(\frac{\mathrm{d}}{\mathrm{d} x}\left(x\left(1+x^{5}\right)^{n}\right)\), or otherwise, show that
\((5 n+1) I_{n}=2^{n}+5 n I_{n-1}\)
(b) Find the exact value of \(I_{3}\).
(a) Find \(\int \sin \theta \cos ^{n} \theta \mathrm{~d} \theta\), where \(n \neq-1\).
Let \(I_{m, n}=\int_{0}^{\frac{1}{2} \pi} \sin ^{m} \theta \cos ^{n} \theta \mathrm{~d} \theta\).
(b) Show that, for \(m \geqslant 2\) and \(n \geqslant 0\),
\(I_{m, n}=\frac{m-1}{m+n} I_{m-2, n}\)
(c) By considering the binomial expansion of \(\left(z+\frac{1}{z}\right)^{5}\), where \(z=\cos \theta+\mathrm{i} \sin \theta\), use de Moivre's theorem to show that
\(\cos ^{5} \theta=a \cos 5 \theta+b \cos 3 \theta+c \cos \theta\)
where \(a\), \(b\) and \(c\) are constants to be determined.
(d) Using the results given in parts (b) and (c), find the exact value of \(I_{2,5}\).
The integral \(I_{n}\), where \(n\) is an integer, is defined by \(I_{n}=\int_{0}^{\frac{1}{2}}\left(1-x^{2}\right)^{-\frac{1}{2} n} \mathrm{~d} x\).
(a) Find the exact value of \(I_{1}\).
(b) By considering \(\frac{\mathrm{d}}{\mathrm{d} x}\left(x\left(1-x^{2}\right)^{-\frac{1}{2} n}\right)\), or otherwise, show that
\[n I_{n+2}=2^{n-1} 3^{-\frac{1}{2} n}+(n-1) I_{n} .\]
(c) Find the exact value of \(I_{5}\) giving the answer in the form \(k \sqrt{3}\), where \(k\) is a rational number to be determined.
Let \(I_{n}=\int_{0}^{1}(1+3 x)^{n} \mathrm{e}^{-3 x} \mathrm{~d} x\), where \(n\) is an integer.
(a) Show that \(3 I_{n}=1-4^{n} \mathrm{e}^{-3}+3 n I_{n-1}\).
(b) Find the exact value of \(I_{2}\).
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 \(I_{5}\).
(a) Starting from the definitions of tanh and sech in terms of exponentials, prove that
\[1-\tanh ^{2} x=\operatorname{sech}^{2} x\]
(b) Using the substitution \(u=\tanh x\), or otherwise, find \(\int \operatorname{sech}^{2} x \tanh ^{2} x \mathrm{~d} x\).
It is given that, for \(n \geqslant 0, I_{n}=\int_{0}^{\ln 3} \operatorname{sech}^{n} x \tanh ^{2} x \mathrm{~d} x\).
(c) Show that, for \(n \geqslant 2\),
\[(n+1) I_{n}=\left(\frac{4}{5}\right)^{3}\left(\frac{3}{5}\right)^{n-2}+(n-2) I_{n-2}\]
[You may use the result that \(\frac{\mathrm{d}}{\mathrm{d} x}(\operatorname{sech} x)=-\tanh x \operatorname{sech} x\).]
(d) Find the value of \(I_{4}\).
Let \(I_n=\int_1^{\sqrt2}(x^2-1)^n\,dx\).
(i) Show that, for \(n\ge1\), \((2n+1)I_n=\sqrt2-2nI_{n-1}\).
(ii) Using the substitution \(x=\sec\theta\), show that \(I_n=\int_0^{\pi/4}\tan^{2n+1}\theta\sec\theta\,d\theta\).
(iii) Deduce the exact value of \(\int_0^{\pi/4}\frac{\sin^7\theta}{\cos^8\theta}\,d\theta\).
Let \(I_{n}=\int_{0}^{\frac{1}{2} \pi} x^{n} \sin x \mathrm{~d} x\).
(i) Prove that, for \(n \geqslant 2\),
\(I_{n}+n(n-1) I_{n-2}=n\left(\frac{1}{2} \pi\right)^{n-1}\)
(ii) Calculate the exact value of \(I_{1}\) and deduce the exact value of \(I_{3}\).
Let \(I_{n}\) denote \(\int_{0}^{2}\left(4+x^{2}\right)^{-n} \mathrm{~d} x\).
(i) Find \(\frac{\mathrm{d}}{\mathrm{d} x}\left(x\left(4+x^{2}\right)^{-n}\right)\) and hence show that
\(8 n I_{n+1}=(2 n-1) I_{n}+2 \times 8^{-n}\)
(ii) Use the result for integrating \(\frac{1}{x^{2}+a^{2}}\) with respect to \(x\), in the List of Formulae (MF10), to find the value of \(I_{1}\) and deduce that
\(I_{3}=\frac{3}{1024} \pi+\frac{1}{128} .\)
Using the substitution \(u=\cos \theta\), or any other method, find \(\int \sin \theta \cos ^{2} \theta \mathrm{~d} \theta\).
It is given that \(I_{n}=\int_{0}^{\frac{1}{2} \pi} \sin ^{n} \theta \cos ^{2} \theta \mathrm{~d} \theta\), for \(n \geqslant 0\). Show that, for \(n \geqslant 2\),
\(I_{n}=\frac{n-1}{n+2} I_{n-2}\)
Hence find the exact value of \(\int_{0}^{\frac{1}{2} \pi} \sin ^{4} \theta \cos ^{2} \theta \mathrm{~d} \theta\).