MISCELLANEOUS EXAMPLES ON CHAPTER VI

1. A function $f(x)$ is defined as being equal to $1+x$ when $x\le 0$, to $x$ when $0<x<1$, to $2–x$ when $1\le x\le 2$, and to $3x–x^2$ when $x>2$. Discuss the continuity of $f(x)$ and the existence and continuity of $f^{\prime }(x)$ for $x=0$, $x=1$, and $x=2$.

 

2. Denoting $a$, $ax+b$, $a{x}^2+2bx+c$, … by $u_0$, $u_1$, $u_2$, …, show that $u_0^2{u}_3–3u_0{u}_1{u}_2+2u_1^3$ and $u_0{u}_4–4u_1{u}_3+3u_2^2$ are independent of $x$.

 

3. If $a_0$, $a_1$, …, $a_{2n}$ are constants and $U_r=(a_0,a_1,\dots ,a_r|x,1{)}^r$, then $$U_0{U}_{2n}–2n{U}_1{U}_{2n-1}+\frac{2n(2n–1)}{1\cdot 2}{U}_2{U}_{2n-2}–\cdots +U_{2n}{U}_0$$ is independent of $x$.

[Differentiate and use the relation $U_r^{\prime }=r{U}_{r-1}$.]

 

4. The first three derivatives of the function $\arcsin (\mu \sin x)–x$, where $\mu >1$, are positive when $0\le x\le \frac{1}{2}\pi$.

 

5. The constituents of a determinant are functions of $x$. Show that its differential coefficient is the sum of the determinants formed by differentiating the constituents of one row only, leaving the rest unaltered.

 

6. If $f_1$, $f_2$, $f_3$, $f_4$ are polynomials of degree not greater than $4$, then $$\left|\begin{array}{cccc}{f}_1& f_2& f_3& f_4\\ f_1^{\prime }& f_2^{\prime }& f_3^{\prime }& f_4^{\prime }\\ f_1”& f_2”& f_3”& f_4”\\ f_1{”}^{\prime }& f_2{”}^{\prime }& f_3{”}^{\prime }& f_4{”}^{\prime }\end{array}\right|$$ is also a polynomial of degree not greater than $4$. [Differentiate five times, using the result of Ex. 5, and rejecting vanishing determinants.]

 

7. If $y^3+3yx+2x^3=0$ then $x^2(1+x^3)y”–\frac{3}{2}x{y}^{\prime }+y=0$.

 

8. Verify that the differential equation $y=\varphi \{\psi (y_1)\}+\varphi \{x–\psi (y_1)\}$, where $y_1$ is the derivative of $y$, and $\psi$ is the function inverse to ${\varphi }^{\prime }$, is satisfied by $y=\varphi (c)+\varphi (x–c)$ or by $y=2\varphi (\frac{1}{2}x)$.

 

9. Verify that the differential equation $y=\{x/\psi (y_1)\}\varphi \{\psi (y_1)\}$, where the notation is the same as that of Ex. 8, is satisfied by $y=c\varphi (x/c)$ or by $y=\beta x$, where $\beta =\varphi (\alpha )/\alpha$ and $\alpha$ is any root of the equation $\varphi (\alpha )–\alpha {\varphi }^{\prime }(\alpha )=0$.

 

10. If $ax+by+c=0$ then $y_2=0$ (suffixes denoting differentiations with respect to $x$). We may express this by saying that the general differential equation of all straight lines is $y_2=0$. Find the general differential equations of (i) all circles with their centres on the axis of $x$, (ii) all parabolas with their axes along the axis of $x$, (iii) all parabolas with their axes parallel to the axis of $y$, (iv) all circles, (v) all parabolas, (vi) all conics.

[The equations are (i) $1+y_1^2+y{y}_2=0$, (ii) $y_1^2+y{y}_2=0$, (iii) $y_3=0$, (iv) $(1+y_1^2)y_3=3y_1{y}_2^2$, (v) $5y_3^2=3y_2{y}_4$, (vi) $9y_2^2{y}_5–45y_2{y}_3{y}_4+40y_3^3=0$. In each case we have only to write down the general equation of the curves in question, and differentiate until we have enough equations to eliminate all the arbitrary constants.]

 

11. Show that the general differential equations of all parabolas and of all conics are respectively $$D_x^2(y_2^{-2/3})=0,D_x^3(y_2^{-2/3})=0.$$

[The equation of a conic may be put in the form $$y=ax+b\pm \sqrt{p{x}^2+2qx+r}.$$ From this we deduce $$y_2=\pm (pr–q^2)/(p{x}^2+2qx+r{)}^{3/2}.$$ If the conic is a parabola then $p=0$.]

 

12. Denoting ${\displaystyle \frac{dy}{dx}}$, ${\displaystyle \frac{1}{2!}}{\displaystyle \frac{d^{2}y}{d{x}^2}}$, ${\displaystyle \frac{1}{3!}}{\displaystyle \frac{d^{3}y}{d{x}^3}}$, ${\displaystyle \frac{1}{4!}}{\displaystyle \frac{d^{4}y}{d{x}^4}}$, … by $t$, $a$, $b$, $c$, … and ${\displaystyle \frac{dx}{dy}}$, ${\displaystyle \frac{1}{2!}}{\displaystyle \frac{d^{2}x}{d{y}^2}}$, ${\displaystyle \frac{1}{3!}}{\displaystyle \frac{d^{3}x}{d{y}^3}}$, ${\displaystyle \frac{1}{4!}}{\displaystyle \frac{d^{4}x}{d{y}^4}}$, … by $\tau$, $\alpha$, $\beta$, $\gamma$, …, show that $$4ac–5b^2=(4\alpha \gamma –5{\beta }^2)/{\tau }^8,bt–a^2=–(\beta \tau –{\alpha }^2)/{\tau }^6.$$ Establish similar formulae for the functions $a^{2}d–3abc–2b^3$, $(1+t^2)b–2a^{2}t$, $2ct–5ab$.

 

13. Prove that, if $y_k$ is the $k$th derivative of $y=\sin (n\arcsin x)$, then $$(1–x^2)y_{k+2}–(2k+1)x{y}_{k+1}+(n^2–k^2)y_k=0.$$

[Prove first when $k=0$, and differentiate $k$ times by Leibniz’ Theorem.]

 

14. Prove the formula $$v{D}_x^{n}u=D_x^n(uv)–n{D}_x^{n-1}(u{D}_{x}v)+\frac{n(n–1)}{1\cdot 2}{D}_x^{n-2}(u{D}_x^{2}v)–\dots$$ where $n$ is any positive integer. [Use the method of induction.]

 

15. A curve is given by $$x=a(2\cos t+\cos 2t),y=a(2\sin t–\sin 2t).$$

Prove (i) that the equations of the tangent and normal, at the point $P$ whose parameter is $t$, are $$x\sin \frac{1}{2}t+y\cos \frac{1}{2}t=a\sin \frac{3}{2}t,x\cos \frac{1}{2}t–y\sin \frac{1}{2}t=3a\cos \frac{3}{2}t;$$ (ii) that the tangent at $P$ meets the curve in the points $Q$, $R$ whose parameters are $-\frac{1}{2}t$ and $\pi –\frac{1}{2}t$; (iii) that $QR=4a$; (iv) that the tangents at $Q$ and $R$ are at right angles and intersect on the circle $x^2+y^2=a^2$; (v) that the normals at $P$, $Q$, and $R$ are concurrent and intersect on the circle $x^2+y^2=9a^2$; (vi) that the equation of the curve is $$(x^2+y^2+12ax+9a^2{)}^2=4a(2x+3a{)}^3.$$

Sketch the form of the curve.

 

16. Show that the equations which define the curve of Ex. 15 may be replaced by $\xi /a=2u+(1/u^2)$, $\eta /a=(2/u)+u^2$, where $\xi =x+yi$, $\eta =x–yi$, $u=\mathrm{Cis}t$. Show that the tangent and normal, at the point defined by $u$, are $$u^2\xi –u\eta =a(u^3–1),u^2\xi +u\eta =3a(u^3+1),$$ and deduce the properties (ii)–(v) of Ex. 15.

 

17. Show that the condition that $x^4+4p{x}^3–4qx–1=0$ should have equal roots may be expressed in the form $(p+q{)}^{2/3}–(p–q{)}^{2/3}=1$.

 

18. The roots of a cubic $f(x)=0$ are $\alpha$, $\beta$, $\gamma$ in ascending order of magnitude. Show that if $[\alpha ,\beta ]$ and $[\beta ,\gamma ]$ are each divided into six equal sub-intervals, then a root of $f^{\prime }(x)=0$ will fall in the fourth interval from $\beta$ on each side. What will be the nature of the cubic in the two cases when a root of $f^{\prime }(x)=0$ falls at a point of division?

 

19. Investigate the maxima and minima of $f(x)$, and the real roots of $f(x)=0$, $f(x)$ being either of the functions $$x–\sin x–\tan \alpha (1–\cos x),x–\sin x–(\alpha –\sin \alpha )–\tan \frac{1}{2}\alpha (\cos \alpha –\cos x),$$ and $\alpha$ an angle between $0$ and $\pi$. Show that in the first case the condition for a double root is that $\tan \alpha –\alpha$ should be a multiple of $\pi$.

 

20. Show that by choice of the ratio $\lambda :\mu$ we can make the roots of $\lambda (a{x}^2+bx+c)+\mu (a^{\prime }{x}^2+b^{\prime }x+c^{\prime })=0$ real and having a difference of any magnitude, unless the roots of the two quadratics are all real and interlace; and that in the excepted case the roots are always real, but there is a lower limit for the magnitude of their difference.

[Consider the form of the graph of the function $(a{x}^2+bx+c)/(a^{\prime }{x}^2+b^{\prime }x+c^{\prime })$: cf. Exs. XLVI. 12 et seq.]

 

21. Prove that $$\pi <\frac{\sin \pi x}{x(1–x)}\le 4$$ when $0<x<1$, and draw the graph of the function.

 

22. Draw the graph of the function $$\pi \cot \pi x–\frac{1}{x}–\frac{1}{x–1}.$$

 

23. Sketch the general form of the graph of $y$, given that $$\frac{dy}{dx}=\frac{(6x^2+x–1)(x–1{)}^2(x+1{)}^3}{x^2}.$$

 

24. A sheet of paper is folded over so that one corner just reaches the opposite side. Show how the paper must be folded to make the length of the crease a maximum.

 

25. The greatest acute angle at which the ellipse $(x^2/a^2)+(y^2/b^2)=1$ can be cut by a concentric circle is $\arctan \{(a^2–b^2)/2ab\}$.

 

26. In a triangle the area $\mathrm{\Delta }$ and the semi-perimeter $s$ are fixed. Show that any maximum or minimum of one of the sides is a root of the equation $s(x–s)x^2+4{\mathrm{\Delta }}^2=0$. Discuss the reality of the roots of this equation, and whether they correspond to maxima or minima.

[The equations $a+b+c=2s$, $s(s–a)(s–b)(s–c)={\mathrm{\Delta }}^2$ determine $a$ and $b$ as functions of $c$. Differentiate with respect to $c$, and suppose that $da/dc=0$. It will be found that $b=c$, $s–b=s–c=\frac{1}{2}a$, from which we deduce that $s(a–s)a^2+4{\mathrm{\Delta }}^2=0$.

This equation has three real roots if $s^4>27{\mathrm{\Delta }}^2$, and one in the contrary case. In an equilateral triangle (the triangle of minimum perimeter for a given area) $s^4=27{\mathrm{\Delta }}^2$; thus it is impossible that $s^4<27{\mathrm{\Delta }}^2$. Hence the equation in $a$ has three real roots, and, since their sum is positive and their product negative, two roots are positive and the third negative. Of the two positive roots one corresponds to a maximum and one to a minimum.]

 

27. The area of the greatest equilateral triangle which can be drawn with its sides passing through three given points $A$, $B$, $C$ is $$2\mathrm{\Delta }+\frac{a^2+b^2+c^2}{2\sqrt{3}},$$ $a$, $b$, $c$ being the sides and $\mathrm{\Delta }$ the area of $ABC$.

 

28. If $\mathrm{\Delta }$, ${\mathrm{\Delta }}^{\prime }$ are the areas of the two maximum isosceles triangles which can be described with their vertices at the origin and their base angles on the cardioid $r=a(1+\cos \theta )$, then $256\mathrm{\Delta }{\mathrm{\Delta }}^{\prime }=25a^4\sqrt{5}$.

 

29. Find the limiting values which $(x^2–4y+8)/(y^2–6x+3)$ approaches as the point $(x,y)$ on the curve $x^{2}y–4x^2–4xy+y^2+16x–2y–7=0$ approaches the position $(2,3)$.

[If we take $(2,3)$ as a new origin, the equation of the curve becomes ${\xi }^2\eta –{\xi }^2+{\eta }^2=0$, and the function given becomes $({\xi }^2+4\xi –4\eta )/({\eta }^2+6\eta –6\xi )$. If we put $\eta =t\xi$, we obtain $\xi =(1–t^2)/t$, $\eta =1–t^2$. The curve has a loop branching at the origin, which corresponds to the two values $t=-1$ and $t=1$. Expressing the given function in terms of $t$, and making $t$ tend to $-1$ or $1$, we obtain the limiting values $-\frac{3}{2}$, $-\frac{2}{3}$.]

 

30. If $f(x)={\displaystyle \frac{1}{\sin x–\sin a}}–{\displaystyle \frac{1}{(x–a)\cos a}}$, then $$\frac{d}{da}\{\underset{x\to a}{lim}f(x)\}–\underset{x\to a}{lim}{f}^{\prime }(x)=\frac{3}{4}{\sec }^{3}a–\frac{5}{12}\sec a.$$

 

31. Show that if $\varphi (x)=1/(1+x^2)$ then ${\varphi }^n(x)=Q_n(x)/(1+x^2{)}^{n+1}$, where $Q_n(x)$ is a polynomial of degree $n$. Show also that

(i) $Q_{n+1}=(1+x^2)Q_n^{\prime }–2(n+1)x{Q}_n$,

(ii) $Q_{n+2}+2(n+2)x{Q}_{n+1}+(n+2)(n+1)(1+x^2)Q_n=0$,

(iii) $(1+x^2)Q_n”–2nx{Q}_n^{\prime }+n(n+1)Q_n=0$,

(iv) $Q_n=(-1{)}^{n}n!\{(n+1)x^n–{\displaystyle \frac{(n+1)n(n–1)}{3!}}{x}^{n-2}+\dots \}$,

(v) all the roots of $Q_n=0$ are real and separated by those of $Q_{n-1}=0$.

 

32. If $f(x)$, $\varphi (x)$, $\psi (x)$ have derivatives when $a\le x\le b$, then there is a value of $\xi$ lying between $a$ and $b$ and such that $$\left|\begin{array}{ccc}f(a)& \varphi (a)& \psi (a)\\ f(b)& \varphi (b)& \psi (b)\\ f^{\prime }(\xi )& {\varphi }^{\prime }(\xi )& {\psi }^{\prime }(\xi )\end{array}\right|=0.$$

[Consider the function formed by replacing the constituents of the third row by $f(x)$, $\varphi (x)$, $\psi (x)$. This theorem reduces to the Mean Value Theorem (§ 125) when $\varphi (x)=x$ and $\psi (x)=1$.]

 

33. Deduce from Ex. 32 the formula $$\frac{f(b)–f(a)}{\varphi (b)–\varphi (a)}=\frac{f^{\prime }(\xi )}{{\varphi }^{\prime }(\xi )}.$$

 

34. If ${\varphi }^{\prime }(x)\to a$ as $x\to \mathrm{\infty }$, then $\varphi (x)/x\to a$. If ${\varphi }^{\prime }(x)\to \mathrm{\infty }$ then $\varphi (x)\to \mathrm{\infty }$. [Use the formula $\varphi (x)–\varphi (x_0)=(x–x_0){\varphi }^{\prime }(\xi )$, where $x_0<\xi <x$.]

 

35. If $\varphi (x)\to a$ as $x\to \mathrm{\infty }$, then ${\varphi }^{\prime }(x)$ cannot tend to any limit other than zero.

 

36. If $\varphi (x)+{\varphi }^{\prime }(x)\to a$ as $x\to \mathrm{\infty }$, then $\varphi (x)\to a$ and ${\varphi }^{\prime }(x)\to 0$.

[Let $\varphi (x)=a+\psi (x)$, so that $\psi (x)+{\psi }^{\prime }(x)\to 0$. If ${\psi }^{\prime }(x)$ is of constant sign, say positive, for all sufficiently large values of $x$, then $\psi (x)$ steadily increases and must tend to a limit $l$ or to $\mathrm{\infty }$. If $\psi (x)\to \mathrm{\infty }$ then ${\psi }^{\prime }(x)\to -\mathrm{\infty }$, which contradicts our hypothesis. If $\psi (x)\to l$ then ${\psi }^{\prime }(x)\to -l$, and this is impossible (Ex. 35) unless $l=0$. Similarly we may dispose of the case in which ${\psi }^{\prime }(x)$ is ultimately negative. If $\psi (x)$ changes sign for values of $x$ which surpass all limit, then these are the maxima and minima of $\psi (x)$. If $x$ has a large value corresponding to a maximum or minimum of $\psi (x)$, then $\psi (x)+{\psi }^{\prime }(x)$ is small and ${\psi }^{\prime }(x)=0$, so that $\psi (x)$ is small. A fortiori are the other values of $\psi (x)$ small when $x$ is large.

For generalisations of this theorem, and alternative lines of proof, see a paper by the author entitled “Generalisations of a limit theorem of Mr Mercer,” in volume 43 of the Quarterly Journal of Mathematics. The simple proof sketched above was suggested by Prof. E. W. Hobson.]

 

37. Show how to reduce $\int R\{x,\sqrt{\frac{ax+b}{mx+n}},\sqrt{\frac{cx+d}{mx+n}}\}dx$ to the integral of a rational function. [Put $mx+n=1/t$ and use Ex. XLIX. 13.]

 

38. Calculate the integrals: $$\begin{array}{c}\int \frac{dx}{(1+x^2{)}^3},\int \sqrt{\frac{x–1}{x+1}}\frac{dx}{x},\int \frac{xdx}{\sqrt{1+x}–\sqrt[3]{1+x}},\\ \int \sqrt{a^2+\sqrt{b^2+\frac{c}{x}}}dx,\int {\csc }^{3}xdx,\int \frac{5\cos x+6}{2\cos x+\sin x+3}dx,\\ \int \frac{dx}{(2–{\sin }^{2}x)(2+\sin x–{\sin }^{2}x)},\int \frac{\cos x\sin xdx}{{\cos }^{4}x+{\sin }^{4}x},\int \csc x\sqrt{\sec 2x}dx,\\ \int \frac{dx}{\sqrt{(1+\sin x)(2+\sin x)}},\int \frac{x+\sin x}{1+\cos x}dx,\int \mathrm{arcsec}xdx,\int (\arcsin x{)}^{2}dx,\\ \int x\arcsin xdx,\int \frac{x\arcsin x}{\sqrt{1–x^2}}dx,\int \frac{\arcsin x}{x^3}dx,\int \frac{\arcsin x}{(1+x{)}^2}dx,\\ \int \frac{\arctan x}{x^2}dx,\int \frac{\arctan x}{(1+x^2{)}^{3/2}}dx,\int \frac{\log ({\alpha }^2+{\beta }^2{x}^2)}{x^2}dx,\int \frac{\log (\alpha +\beta x)}{(a+bx{)}^2}dx.\end{array}$$

 

39. Formulae of reduction. (i) Show that $$\begin{array}{c}2(n–1)(q–\frac{1}{4}{p}^2)\int \frac{dx}{(x^2+px+q{)}^n}\\ =\frac{x+\frac{1}{2}p}{(x^2+px+q{)}^{n-1}}+(2n–3)\int \frac{dx}{(x^2+px+q{)}^{n-1}}.\end{array}$$

[Put $x+\frac{1}{2}p=t$, $q–\frac{1}{4}{p}^2=\lambda$: then we obtain $$\begin{array}{rl}\int \frac{dt}{(t^2+\lambda {)}^n}& =\frac{1}{\lambda }\int \frac{dt}{(t^2+\lambda {)}^{n-1}}–\frac{1}{\lambda }\int \frac{t^{2}dt}{(t^2+\lambda {)}^n}\\ & =\frac{1}{\lambda }\int \frac{dt}{(t^2+\lambda {)}^{n-1}}+\frac{1}{2\lambda (n-1)}\int t\frac{d}{dt}\left\{\frac{1}{(t^2+\lambda {)}^{n-1}}\right\}dt,\end{array}$$ and the result follows on integrating by parts.

A formula such as this is called a formula of reduction. It is most useful when $n$ is a positive integer. We can then express $\int \frac{dx}{(x^2+px+q{)}^n}$ in terms of $\int \frac{dx}{(x^2+px+q{)}^{n-1}}$, and so evaluate the integral for every value of $n$ in turn.]

(ii) Show that if $I_{p,q}=\int x^p(1+x{)}^{q}dx$ then $$(p+1)I_{p,q}=x^{p+1}(1+x{)}^q–q{I}_{p+1,q-1},$$ and obtain a similar formula connecting $I_{p,q}$ with $I_{p-1,q+1}$. Show also, by means of the substitution $x=-y/(1+y)$, that $$I_{p,q}=(-1{)}^{p+1}\int y^p(1+y{)}^{-p-q-2}dy.$$

(iii) Show that if $X=a+bx$ then $$\begin{array}{rl}\int x{X}^{-1/3}dx& =-3(3a–2bx)X^{2/3}/10b^2,\\ \int x^2{X}^{-1/3}dx& =3(9a^2–6abx+5b^2{x}^2)X^{2/3}/40b^3,\\ \int x{X}^{-1/4}dx& =-4(4a–3bx)X^{3/4}/21b^2,\\ \int x^2{X}^{-1/4}dx& =4(32a^2–24abx+21b^2{x}^2)X^{3/4}/231b^3.\end{array}$$

(iv) If $I_{m,n}=\int \frac{x^{m}dx}{(1+x^2{)}^n}$ then $$2(n–1)I_{m,n}=-x^{m-1}(1+x^2{)}^{-(n-1)}+(m–1)I_{m-2,n-1}.$$

(v) If $I_n=\int x^n\cos \beta xdx$ and $J_n=\int x^n\sin \beta xdx$ then $$\beta I_n=x^n\sin \beta x–n{J}_{n-1},\beta J_n=-x^n\cos \beta x+n{I}_{n-1}.$$

(vi) If $I_n=\int {\cos }^{n}xdx$ and $J_n=\int {\sin }^{n}xdx$ then $$n{I}_n=\sin x{\cos }^{n-1}x+(n–1)I_{n-2},n{J}_n=-\cos x{\sin }^{n-1}x+(n–1)J_{n-2}.$$

(vii) If $I_n=\int {\tan }^{n}xdx$ then $(n–1)(I_n+I_{n-2})={\tan }^{n-1}x$.

(viii) If $I_{m,n}=\int {\cos }^{m}x{\sin }^{n}xdx$ then $$\begin{array}{rlrl}2(m+n)I_{m,n}& =-& & {\cos }^{m+1}x{\sin }^{n-1}x+(n–1)I_{m,n-2}\\ & =& & {\cos }^{m-1}x{\sin }^{n+1}x+(m–1)I_{m-2,n}.\end{array}$$

[We have $$\begin{array}{rl}(m+1)I_{m,n}& =-\int {\sin }^{n-1}x\frac{d}{dx}({\cos }^{m+1}x)dx\\ & =-{\cos }^{m+1}x{\sin }^{n-1}x+(n–1)\int {\cos }^{m+2}x{\sin }^{n-2}xdx\\ & =-{\cos }^{m+1}x{\sin }^{n-1}x+(n–1)(I_{m,n-2}–I_{m,n}),\end{array}$$ which leads to the first reduction formula.]

(ix) Connect $I_{m,n}=\int {\sin }^{m}x\sin nxdx$ with $I_{m-2,n}$.

(x) If $I_{m,n}=\int x^m{\csc }^{n}xdx$ then $$\begin{array}{c}(n–1)(n–2)I_{m,n}=(n–2{)}^2{I}_{m,n-2}+m(m–1)I_{m-2,n-2}\\ -x^{m-1}{\csc }^{n-1}x\{m\sin x+(n–2)x\cos x\}.\end{array}$$

(xi) If $I_n=\int (a+b\cos x{)}^{-n}dx$ then $$(n–1)(a^2–b^2)I_n=-b\sin x(a+b\cos x{)}^{-(n-1)}+(2n–3)a{I}_{n-1}–(n–2)I_{n-2}.$$

(xii) If $I_n=\int (a{\cos }^{2}x+2h\cos x\sin x+b{\sin }^{2}x{)}^{-n}dx$ then $$4n(n+1)(ab–h^2)I_{n+2}–2n(2n+1)(a+b)I_{n+1}+4n^2{I}_n=-\frac{d^2{I}_n}{d{x}^2}.$$

If $I_{m,n}=\int x^m(\log x{)}^{n}dx$ then $$(m+1)I_{m,n}=x^{m+1}(\log x{)}^n–n{I}_{m,n-1}.$$

 

40. If $n$ is a positive integer then the value of $\int x^m(\log x{)}^{n}dx$ is $$x^{m+1}\{\frac{(\log x{)}^n}{m+1}–\frac{n(\log x{)}^{n-1}}{(m+1{)}^2}+\frac{n(n–1)(\log x{)}^{n-2}}{(m+1{)}^3}–\cdots +\frac{(-1{)}^{n}n!}{(m+1{)}^{n+1}}\}.$$

 

41. Show that the most general function $\varphi (x)$, such that $\varphi ”+a^2\varphi =0$ for all values of $x$, may be expressed in either of the forms $A\cos ax+B\sin ax$, $\rho \cos (ax+ϵ)$, where $A$, $B$, $\rho$, $ϵ$ are constants. [Multiplying by $2{\varphi }^{\prime }$ and integrating we obtain ${\varphi }^{\prime 2}+a^2{\varphi }^2=a^2{b}^2$, where $b$ is a constant, from which we deduce that $ax=\int \frac{d\varphi }{\sqrt{b^2–{\varphi }^2}}$.]

 

42. Determine the most general functions $y$ and $z$ such that $y^{\prime }+\omega z=0$, and $z^{\prime }–\omega y=0$, where $\omega$ is a constant and dashes denote differentiation with respect to $x$.

 

43. The area of the curve given by $$x=\cos \varphi +\frac{\sin \alpha \sin \varphi }{1–{\cos }^2\alpha {\sin }^2\varphi },y=\sin \varphi –\frac{\sin \alpha \cos \varphi }{1–{\cos }^2\alpha {\sin }^2\varphi },$$ where $\alpha$ is a positive acute angle, is $\frac{1}{2}\pi (1+\sin \alpha {)}^2/\sin \alpha$.

 

44. The projection of a chord of a circle of radius $a$ on a diameter is of constant length $2a\cos \beta$; show that the locus of the middle point of the chord consists of two loops, and that the area of either is $a^2(\beta –\cos \beta \sin \beta )$.

 

45. Show that the length of a quadrant of the curve $(x/a{)}^{2/3}+(y/b{)}^{2/3}=1$ is $(a^2+ab+b^2)/(a+b)$.

 

46. A point $A$ is inside a circle of radius $a$, at a distance $b$ from the centre. Show that the locus of the foot of the perpendicular drawn from $A$ to a tangent to the circle encloses an area $\pi (a^2+\frac{1}{2}{b}^2)$.

 

47. Prove that if $(a,b,c,f,g,h|x,y,1{)}^2=0$ is the equation of a conic, then $$\int \frac{dx}{(lx+my+n)(hx+by+f)}=\alpha \log \frac{PT}{P{T}^{\prime }}+\beta ,$$ where $PT$, $P{T}^{\prime }$ are the perpendiculars from a point $P$ of the conic on the tangents at the ends of the chord $lx+my+n=0$, and $\alpha$, $\beta$ are constants.

 

48. Show that $$\int \frac{a{x}^2+2bx+c}{(A{x}^2+2Bx+C{)}^2}dx$$ will be a rational function of $x$ if and only if one or other of $AC–B^2$ and $aC+cA–2bB$ is zero.¹

 

49. Show that the necessary and sufficient condition that $$\int \frac{f(x)}{\{F(x){\}}^2}dx,$$ where $f$ and $F$ are polynomials of which the latter has no repeated factor, should be a rational function of $x$, is that $f^{\prime }{F}^{\prime }–fF”$ should be divisible by $F$.

 

50. Show that $$\int \frac{\alpha \cos x+\beta \sin x+\gamma }{(1–e\cos x{)}^2}dx$$ is a rational function of $\cos x$ and $\sin x$ if and only if $\alpha e+\gamma =0$; and determine the integral when this condition is satisfied.

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1. See the author’s tract quoted on § 131.↩︎

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$\leftarrow$146. Lengths of plane curves

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