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

2. Denoting $$a$$, $$ax + b$$, $$ax^{2} + 2bx + c$$, … 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 $U_{0}U_{2n} – 2nU_{1}U_{2n-1} + \frac{2n(2n – 1)}{1\cdot2} U_{2}U_{2n-2} – \dots + U_{2n}U_{0}$ is independent of $$x$$.

[Differentiate and use the relation $$U_{r}’ = rU_{r-1}$$.]

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

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 $\begin{vmatrix} f_{1}& f_{2}& f_{3}& f_{4}\\ f_{1}’& f_{2}’& f_{3}’& f_{4}’\\ f_{1}”& f_{2}”& f_{3}”& f_{4}”\\ f_{1}”’& f_{2}”’& f_{3}”’& f_{4}”’ \end{vmatrix}$ 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 $$x^{2}(1 + x^{3})y” – \frac{3}{2}xy’ + y = 0$$.

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

9. Verify that the differential equation $$y = \{x/\psi(y_{1})\} \phi\{\psi(y_{1})\}$$, where the notation is the same as that of Ex. 8, is satisfied by $$y = c\phi(x/c)$$ or by $$y = \beta x$$, where $$\beta = \phi(\alpha)/\alpha$$ and $$\alpha$$ is any root of the equation $$\phi(\alpha) – \alpha\phi'(\alpha) = 0$$.

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) $$1 + y_{1}^{2} + yy_{2} = 0$$, (ii) $$y_{1}^{2} + yy_{2} = 0$$, (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 $D_{x}^{2} (y_{2}^{-2/3}) = 0,\quad D_{x}^{3} (y_{2}^{-2/3}) = 0.$

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

12. Denoting $$\dfrac{dy}{dx}$$, $$\dfrac{1}{2!}\, \dfrac{d^{2}y}{dx^{2}}$$, $$\dfrac{1}{3!}\, \dfrac{d^{3}y}{dx^{3}}$$, $$\dfrac{1}{4!}\, \dfrac{d^{4}y}{dx^{4}}$$, … by $$t$$, $$a$$, $$b$$, $$c$$, … and $$\dfrac{dx}{dy}$$, $$\dfrac{1}{2!}\, \dfrac{d^{2}x}{dy^{2}}$$, $$\dfrac{1}{3!}\, \dfrac{d^{3}x}{dy^{3}}$$, $$\dfrac{1}{4!}\, \dfrac{d^{4}x}{dy^{4}}$$, … by $$\tau$$, $$\alpha$$, $$\beta$$, $$\gamma$$, …, show that $4ac – 5b^{2} = (4\alpha\gamma – 5\beta^{2})/\tau^{8},\quad bt – a^{2} = – (\beta\tau – \alpha^{2})/\tau^{6}.$ 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 $(1 – x^{2})y_{k+2} – (2k + 1)xy_{k+1} + (n^{2} – k^{2})y_{k} = 0.$

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

14. Prove the formula $vD_{x}^{n}u = D_{x}^{n}(uv) – nD_{x}^{n-1}(uD_{x}v) + \frac{n(n – 1)}{1\cdot2} D_{x}^{n-2}(uD_{x}^{2}v) – \dots$ where $$n$$ is any positive integer. [Use the method of induction.]

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

Prove (i) that the equations of the tangent and normal, at the point $$P$$ whose parameter is $$t$$, are $x\sin \tfrac{1}{2} t + y\cos \tfrac{1}{2} t = a\sin \tfrac{3}{2} t,\quad x\cos \tfrac{1}{2} t – y\sin \tfrac{1}{2} t = 3a\cos \tfrac{3}{2} t;$ (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 $$\xi = x + yi$$, $$\eta = x – yi$$, $$u = \operatorname{Cis} t$$. Show that the tangent and normal, at the point defined by $$u$$, are $u^{2}\xi – u\eta = a(u^{3} – 1),\quad u^{2}\xi + u\eta = 3a(u^{3} + 1),$ and deduce the properties (ii)–(v) of Ex. 15.

17. Show that the condition that $$x^{4} + 4px^{3} – 4qx – 1 = 0$$ 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 $$f'(x) = 0$$ 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 $$f'(x) = 0$$ 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 $x – \sin x – \tan\alpha (1 – \cos x),\quad x – \sin x – (\alpha – \sin\alpha) – \tan \tfrac{1}{2}\alpha (\cos\alpha – \cos x),$ 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 $$\lambda(ax^{2} + bx + c) + \mu(a’x^{2} + b’x + c’) = 0$$ 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 $$(ax^{2} + bx + c)/(a’x^{2} + b’x + c’)$$: cf. Exs. XLVI. 12 et seq.]

21. Prove that $\pi < \frac{\sin \pi x}{x(1 – x)} \leq 4$ 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 $$\Delta$$ and the semi-perimeter $$s$$ are fixed. Show that any maximum or minimum of one of the sides is a root of the equation $$s(x – s) x^{2} + 4\Delta^{2} = 0$$. Discuss the reality of the roots of this equation, and whether they correspond to maxima or minima.

[The equations $$a + b + c = 2s$$, $$s(s – a)(s – b)(s – c) = \Delta^{2}$$ 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 $$s(a – s)a^{2} + 4\Delta^{2} = 0$$.

This equation has three real roots if $$s^{4} > 27\Delta^{2}$$, and one in the contrary case. In an equilateral triangle (the triangle of minimum perimeter for a given area) $$s^{4} = 27\Delta^{2}$$; thus it is impossible that $$s^{4} < 27\Delta^{2}$$. 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 $2\Delta + \frac{a^{2} + b^{2} + c^{2}}{2\sqrt{3}},$ $$a$$, $$b$$, $$c$$ being the sides and $$\Delta$$ the area of $$ABC$$.

28. If $$\Delta$$, $$\Delta’$$ 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 $$256\Delta\Delta’ = 25a^{4}\sqrt{5}$$.

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 $$f(x) = \dfrac{1}{\sin x – \sin a} – \dfrac{1}{(x – a)\cos a}$$, then $\frac{d}{da}\{\lim_{x \to a} f(x)\} – \lim_{x \to a}f'(x) = \tfrac{3}{4} \sec^{3} a – \tfrac{5}{12} \sec a.$

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

(i) $$Q_{n+1} = (1 + x^{2}) Q_{n}’ – 2(n + 1) x Q_{n}$$,

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

(iii) $$(1 + x^{2}) Q_{n}” – 2nx Q_{n}’ + n(n + 1)Q_{n} = 0$$,

(iv) $$Q_{n} = (-1)^{n} n!\left\{(n + 1)x^{n} – \dfrac{(n + 1)n(n – 1)}{3!} x^{n-2} + \dots\right\}$$,

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

32. If $$f(x)$$, $$\phi(x)$$, $$\psi(x)$$ have derivatives when $$a \leq x \leq b$$, then there is a value of $$\xi$$ lying between $$a$$ and $$b$$ and such that $\begin{vmatrix} f(a) & \phi(a) & \psi(a)\\ f(b) & \phi(b) & \psi(b)\\ f'(\xi)& \phi'(\xi)& \psi'(\xi) \end{vmatrix} =0.$

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

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

34. If $$\phi'(x) \to a$$ as $$x \to \infty$$, then $$\phi(x)/x \to a$$. If $$\phi'(x) \to \infty$$ then $$\phi(x) \to \infty$$. [Use the formula $$\phi(x) – \phi(x_{0}) = (x – x_{0})\phi'(\xi)$$, where $$x_{0} < \xi < x$$.]

35. If $$\phi(x) \to a$$ as $$x \to \infty$$, then $$\phi'(x)$$ cannot tend to any limit other than zero.

36. If $$\phi(x) + \phi'(x) \to a$$ as $$x \to \infty$$, then $$\phi(x) \to a$$ and $$\phi'(x) \to 0$$.

[Let $$\phi(x) = a + \psi(x)$$, so that $$\psi(x) + \psi'(x) \to 0$$. If $$\psi'(x)$$ 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 $$\infty$$. If $$\psi(x) \to \infty$$ then $$\psi'(x) \to -\infty$$, which contradicts our hypothesis. If $$\psi(x) \to l$$ then $$\psi'(x) \to -l$$, and this is impossible (Ex. 35) unless $$l = 0$$. Similarly we may dispose of the case in which $$\psi'(x)$$ 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 $$\psi(x) + \psi'(x)$$ is small and $$\psi'(x) = 0$$, 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 $$\int R\left\{x, \sqrt{\frac{ax + b}{mx + n}}, \sqrt{\frac{cx + d}{mx + n}}\right\} dx$$ to the integral of a rational function. [Put $$mx + n = 1/t$$ and use Ex. XLIX. 13.]

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

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

[Put $$x + \frac{1}{2}p = t$$, $$q – \frac{1}{4}p^{2} = \lambda$$: then we obtain \begin{aligned} \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{aligned} 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 $(p + 1) I_{p, q} = x^{p+1}(1 + x)^{q} – qI_{p+1, q-1},$ 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{aligned} \int xX^{-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 xX^{-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{aligned}

(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 $$I_{n} = \int x^{n} \cos\beta x\, dx$$ and $$J_{n} = \int x^{n} \sin\beta x\, dx$$ then $\beta I_{n} = x^{n} \sin\beta x – nJ_{n-1},\quad \beta J_{n} = -x^{n} \cos\beta x + nI_{n-1}.$

(vi) If $$I_{n} = \int \cos^{n} x\, dx$$ and $$J_{n} = \int \sin^{n} x\, dx$$ then $nI_{n} = \sin x\cos^{n-1} x + (n – 1) I_{n-2},\quad nJ_{n} = -\cos x\sin^{n-1} x + (n – 1) J_{n-2}.$

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

(viii) If $$I_{m, n} = \int \cos^{m}x \sin^{n}x\, dx$$ then \begin{aligned} {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{aligned}

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

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

(x) If $$I_{m, n} = \int x^{m} \csc^{n}x\, dx$$ then $\begin{gathered} (n – 1)(n – 2)I_{m, n} = (n – 2)^{2}I_{m, n-2} + m(m – 1)I_{m-2, n-2}\\ -x^{m-1} \csc^{n-1}x \{m\sin x + (n – 2) x\cos x\}.\end{gathered}$

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

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

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

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

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

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

43. The area of the curve given by $x = \cos\phi + \frac{\sin\alpha \sin\phi}{1 – \cos^{2}\alpha \sin^{2}\phi},\quad y = \sin\phi – \frac{\sin\alpha \cos\phi}{1 – \cos^{2}\alpha \sin^{2}\phi},$ 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 $\int \frac{dx}{(lx + my + n)(hx + by + f)} = \alpha\log \frac{PT}{PT’} + \beta,$ where $$PT$$, $$PT’$$ 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 $\int \frac{ax^{2} + 2bx + c}{(Ax^{2} + 2Bx + C)^{2}}\, dx$ 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 $$f’F’ – fF”$$ 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.

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