MISCELLANEOUS EXAMPLES ON CHAPTER IX

1. Given that ${\log }_{10}e=.4343$ and that $2^{10}$ and $3^{21}$ are nearly equal to powers of $10$, calculate ${\log }_{10}2$ and ${\log }_{10}3$ to four places of decimals.

 

2. Determine which of $(\frac{1}{2}e{)}^{\sqrt{3}}$ and $(\sqrt{2}{)}^{\frac{1}{2}\pi }$ is the greater. [Take logarithms and observe that $\sqrt{3}/(\sqrt{3}+\frac{1}{4}\pi )<\frac{2}{5}\sqrt{3}<.6929<\log 2$.]

 

3. Show that ${\log }_{10}n$ cannot be a rational number if $n$ is any positive integer not a power of $10$. [If $n$ is not divisible by $10$, and ${\log }_{10}n=p/q$, we have ${10}^p=n^q$, which is impossible, since ${10}^p$ ends with $0$ and $n^q$ does not. If $n={10}^{a}N$, where $N$ is not divisible by $10$, then ${\log }_{10}N$ and therefore $${\log }_{10}n=a+{\log }_{10}N$$ cannot be rational.]

 

4. For what values of $x$ are the functions $\log x$, $\log \log x$, $\log \log \log x$, … (a) equal to $0$ (b) equal to $1$ (c) not defined? Consider also the same question for the functions $lx$, $llx$, $lllx$, …, where $lx=\log |x|$.

 

5. Show that $$\log x–(\genfrac{}{}{0ex}{}{n}{1})\log (x+1)+(\genfrac{}{}{0ex}{}{n}{2})\log (x+2)–\cdots +(-1{)}^n\log (x+n)$$ is negative and increases steadily towards $0$ as $x$ increases from $0$ towards $\mathrm{\infty }$.

[The derivative of the function is $$\sum _0^n(-1{)}^r(\genfrac{}{}{0ex}{}{n}{r})\frac{1}{x+r}=\frac{n!}{x(x+1)\dots (x+n)},$$ as is easily seen by splitting up the right-hand side into partial fractions. This expression is positive, and the function itself tends to zero as $x\to \mathrm{\infty }$, since $$\log (x+r)=\log x+{ϵ}_x,$$ where ${ϵ}_x\to 0$, and $1–{\displaystyle (\genfrac{}{}{0ex}{}{n}{1})}+{\displaystyle (\genfrac{}{}{0ex}{}{n}{2})}–\cdots =0$.]

 

6. Prove that $${\left(\frac{d}{dx}\right)}^n\frac{\log x}{x}=\frac{(-1{)}^{n}n!}{x^{n+1}}(\log x–1–\frac{1}{2}–\cdots –\frac{1}{n}).$$

 

7. If $x>-1$ then $x^2>(1+x)\{\log (1+x){\}}^2$.

[Put $1+x=e^{\xi }$, and use the fact that $\sinh \xi >\xi$ when $\xi >0$.]

 

8. Show that $\{\log (1+x)\}/x$ and $x/\{(1+x)\log (1+x)\}$ both decrease steadily as $x$ increases from $0$ towards $\mathrm{\infty }$.

 

9. Show that, as $x$ increases from $-1$ towards $\mathrm{\infty }$, the function $(1+x{)}^{-1/x}$ assumes once and only once every value between $0$ and $1$.

 

10. Show that ${\displaystyle \frac{1}{\log (1+x)}}–{\displaystyle \frac{1}{x}}\to {\displaystyle \frac{1}{2}}$ as $x\to 0$.

 

11. Show that ${\displaystyle \frac{1}{\log (1+x)}}–{\displaystyle \frac{1}{x}}$ decreases steadily from $1$ to $0$ as $x$ increases from $-1$ towards $\mathrm{\infty }$. [The function is undefined when $x=0$, but if we attribute to it the value $\frac{1}{2}$ when $x=0$ it becomes continuous for $x=0$. Use Ex. 7 to show that the derivative is negative.]

 

12. Show that the function $(\log \xi –\log x)/(\xi –x)$, where $\xi$ is positive, decreases steadily as $x$ increases from $0$ to $\xi$, and find its limit as $x\to \xi$.

 

13. Show that $e^x>M{x}^N$, where $M$ and $N$ are large positive numbers, if $x$ is greater than the greater of $2\log M$ and $16N^2$.

[It is easy to prove that $\log x<2\sqrt{x}$; and so the inequality given is certainly satisfied if $$x>\log M+2N\sqrt{x},$$ and therefore certainly satisfied if $\frac{1}{2}x>\log M$, $\frac{1}{2}x>2N\sqrt{x}$.]

 

14. If $f(x)$ and $\varphi (x)$ tend to infinity as $x\to \mathrm{\infty }$, and $f^{\prime }(x)/{\varphi }^{\prime }(x)\to \mathrm{\infty }$, then $f(x)/\varphi (x)\to \mathrm{\infty }$. [Use the result of Ch. VI, Misc. Ex. 33.] By taking $f(x)=x^{\alpha }$, $\varphi (x)=\log x$, prove that $(\log x)/x^{\alpha }\to 0$ for all positive values of $\alpha$.

 

15. If $p$ and $q$ are positive integers then $$\frac{1}{pn+1}+\frac{1}{pn+2}+\cdots +\frac{1}{qn}\to \log \left(\frac{q}{p}\right)$$ as $n\to \mathrm{\infty }$. [Cf. Ex. LXXVIII. 6.]

 

16. Prove that if $x$ is positive then $n\log \{\frac{1}{2}(1+x^{1/n})\}\to -\frac{1}{2}\log x$ as $n\to \mathrm{\infty }$. [We have $$n\log \{\frac{1}{2}(1+x^{1/n})\}=n\log \{1–\frac{1}{2}(1–x^{1/n})\}=\frac{1}{2}n(1–x^{1/n})\frac{\log (1–u)}{u}$$ where $u=\frac{1}{2}(1–x^{1/n})$. Now use § 209 and Ex. LXXXII. 4.]

 

17. Prove that if $a$ and $b$ are positive then $$\{\frac{1}{2}(a^{1/n}+b^{1/n}){\}}^n\to \sqrt{ab}.$$

[Take logarithms and use Ex. 16.]

 

18. Show that $$1+\frac{1}{3}+\frac{1}{5}+\cdots +\frac{1}{2n–1}=\frac{1}{2}\log n+\log 2+\frac{1}{2}\gamma +{ϵ}_n,$$ where $\gamma$ is Euler’s constant (Ex. LXXXIX. 1) and ${ϵ}_n\to 0$ as $n\to \mathrm{\infty }$.

 

19. Show that $$1+\frac{1}{3}–\frac{1}{2}+\frac{1}{5}+\frac{1}{7}–\frac{1}{4}+\frac{1}{9}+\cdots =\frac{3}{2}\log 2,$$ the series being formed from the series $1–\frac{1}{2}+\frac{1}{3}–\dots$ by taking alternately two positive terms and then one negative. [The sum of the first $3n$ terms is $$\begin{array}{c}1+\frac{1}{3}+\frac{1}{5}+\cdots +\frac{1}{4n–1}–\frac{1}{2}(1+\frac{1}{2}+\cdots +\frac{1}{n})\\ =\frac{1}{2}\log 2n+\log 2+\frac{1}{2}\gamma +{ϵ}_n–\frac{1}{2}(\log n+\gamma +{ϵ}_n^{\prime }),\end{array}$$ where ${ϵ}_n$ and ${ϵ}_n^{\prime }$ tend to $0$ as $n\to \mathrm{\infty }$. (Cf. Ex. LXXVIII. 6).]

 

20. Show that $1–\frac{1}{2}–\frac{1}{4}+\frac{1}{3}–\frac{1}{6}–\frac{1}{8}+\frac{1}{5}–\frac{1}{10}–\cdots =\frac{1}{2}\log 2$.

 

21. Prove that $$\sum _1^n\frac{1}{\nu (36{\nu }^2–1)}=-3+3{\mathrm{\Sigma }}_{3n+1}–{\mathrm{\Sigma }}_n–S_n$$ where $S_n=1+{\displaystyle \frac{1}{2}}+\cdots +{\displaystyle \frac{1}{n}}$, ${\mathrm{\Sigma }}_n=1+{\displaystyle \frac{1}{3}}+\cdots +{\displaystyle \frac{1}{2n–1}}$. Hence prove that the sum of the series when continued to infinity is $$-3+\frac{3}{2}\log 3+2\log 2.$$

 

22. Show that $$\sum _1^{\mathrm{\infty }}\frac{1}{n(4n^2–1)}=2\log 2–1,\sum _1^{\mathrm{\infty }}\frac{1}{n(9n^2–1)}=\frac{3}{2}(\log 3–1).$$

 

23. Prove that the sums of the four series $$\sum _1^{\mathrm{\infty }}\frac{1}{4n^2–1},\sum _1^{\mathrm{\infty }}\frac{(-1{)}^{n-1}}{4n^2–1},\sum _1^{\mathrm{\infty }}\frac{1}{(2n+1{)}^2–1},\sum _1^{\mathrm{\infty }}\frac{(-1{)}^{n-1}}{(2n+1{)}^2–1}$$ are $\frac{1}{2}$, $\frac{1}{4}\pi –\frac{1}{2}$, $\frac{1}{4}$, $\frac{1}{2}\log 2–\frac{1}{4}$ respectively.

 

24. Prove that $n!(a/n{)}^n$ tends to $0$ or to $\mathrm{\infty }$ according as $a<e$ or $a>e$.

[If $u_n=n!(a/n{)}^n$ then $u_{n+1}/u_n=a\{1+(1/n){\}}^{-n}\to a/e$. It can be shown that the function tends to $\mathrm{\infty }$ when $a=e$: for a proof, which is rather beyond the scope of the theorems of this chapter, see Bromwich’s Infinite Series, pp. 461 et seq.]

 

25. Find the limit as $x\to \mathrm{\infty }$ of $${\left(\frac{a_0+a_{1}x+\cdots +a_r{x}^r}{b_0+b_{1}x+\cdots +b_r{x}^r}\right)}^{{\lambda }_0+{\lambda }_{1}x},$$ distinguishing the different cases which may arise.

 

26. Prove that $$\sum \log (1+\frac{x}{n})(x>0)$$ diverges to $\mathrm{\infty }$. [Compare with $\sum (x/n)$.] Deduce that if $x$ is positive then $$(1+x)(2+x)\dots (n+x)/n!\to \mathrm{\infty }$$ as $n\to \mathrm{\infty }$. [The logarithm of the function is $\sum _1^n\log (1+{\displaystyle \frac{x}{\nu }})$.]

 

27. Prove that if $x>-1$ then $$\begin{array}{c}\frac{1}{(x+1{)}^2}=\frac{1}{(x+1)(x+2)}+\frac{1!}{(x+1)(x+2)(x+3)}\\ +\frac{2!}{(x+1)(x+2)(x+3)(x+4)}+\dots .\end{array}$$

[The difference between $1/(x+1{)}^2$ and the sum of the first $n$ terms of the series is $$\frac{1}{(x+1{)}^2}\frac{n!}{(x+2)(x+3)\dots (x+n+1)}.]$$

 

28. No equation of the type $$A{e}^{\alpha x}+B{e}^{\beta x}+\cdots =0,$$ where $A$, $B$, … are polynomials and $\alpha$, $\beta$, … different real numbers, can hold for all values of $x$. [If $\alpha$ is the algebraically greatest of $\alpha$, $\beta$, …, then the term $A{e}^{\alpha x}$ outweighs all the rest as $x\to \mathrm{\infty }$.]

 

29. Show that the sequence $$a_1=e,a_2=e^{e^2},a_3=e^{e^{e^3}},\dots$$ tends to infinity more rapidly than any member of the exponential scale.

[Let $e_1(x)=e^x$, $e_2(x)=e^{e_1(x)}$, and so on. Then, if $e_k(x)$ is any member of the exponential scale, $a_n>e_k(n)$ when $n>k$.]

 

30. Prove that $$\frac{d}{dx}\{\varphi (x){\}}^{\psi (x)}=\frac{d}{dx}\{\varphi (x){\}}^{\alpha }+\frac{d}{dx}\{{\beta }^{\psi (x)}\}$$ where $\alpha$ is to be put equal to $\psi (x)$ and $\beta$ to $\varphi (x)$ after differentiation. Establish a similar rule for the differentiation of $\varphi (x{)}^{[\{\psi (x){\}}^{\chi (x)}]}$.

 

31. Prove that if $D_x^n{e}^{-x^2}=e^{-x^2}{\varphi }_n(x)$ then (i) ${\varphi }_n(x)$ is a polynomial of degree $n$, (ii) ${\varphi }_{n+1}=-2x{\varphi }_n+{\varphi }_n^{\prime }$, and (iii) all the roots of ${\varphi }_n=0$ are real and distinct, and separated by those of ${\varphi }_{n-1}=0$. [To prove (iii) assume the truth of the result for $\kappa =1$, $2$, …, $n$, and consider the signs of ${\varphi }_{n+1}$ for the $n$ values of $x$ for which ${\varphi }_n=0$ and for large (positive or negative) values of $x$.]

 

32. The general solution of $f(xy)=f(x)f(y)$, where $f$ is a differentiable function, is $x^a$, where $a$ is a constant: and that of $$f(x+y)+f(x–y)=2f(x)f(y)$$ is $\cosh ax$ or $\cos ax$, according as $f”(0)$ is positive or negative. [In proving the second result assume that $f$ has derivatives of the first three orders. Then $$2f(x)+y^2\{f”(x)+{ϵ}_y\}=2f(x)[f(0)+y{f}^{\prime }(0)+\frac{1}{2}{y}^2\{f”(0)+{ϵ}_y^{\prime }\}],$$ where ${ϵ}_y$ and ${ϵ}_y^{\prime }$ tend to zero with $y$. It follows that $f(0)=1$, $f^{\prime }(0)=0$, $f”(x)=f”(0)f(x)$, so that $a=\sqrt{f”(0)}$ or $a=\sqrt{-f”(0)}$.]

 

33. How do the functions $x^{\sin (1/x)}$, $x^{{\sin }^2(1/x)}$, $x^{\csc (1/x)}$ behave as $x\to +0$?

 

34. Trace the curves $y=\tan x{e}^{\tan x}$, $y=\sin x\log \tan \frac{1}{2}x$.

 

35. The equation $e^x=ax+b$ has one real root if $a<0$ or $a=0$, $b>0$. If $a>0$ then it has two real roots or none, according as $a\log a>b–a$ or $a\log a<b–a$.

 

36. Show by graphical considerations that the equation $e^x=a{x}^2+2bx+c$ has one, two, or three real roots if $a>0$, none, one, or two if $a<0$; and show how to distinguish between the different cases.

 

37. Trace the curve $y={\displaystyle \frac{1}{x}}\log \left({\displaystyle \frac{e^x–1}{x}}\right)$, showing that the point $(0,\frac{1}{2})$ is a centre of symmetry, and that as $x$ increases through all real values, $y$ steadily increases from $0$ to $1$. Deduce that the equation $$\frac{1}{x}\log \left(\frac{e^x–1}{x}\right)=\alpha$$ has no real root unless $0<\alpha <1$, and then one, whose sign is the same as that of $\alpha –\frac{1}{2}$. [In the first place $$y–\frac{1}{2}=\frac{1}{x}\{\log \left(\frac{e^x–1}{x}\right)–\log e^{\frac{1}{2}x}\}=\frac{1}{x}\log \left(\frac{\sinh \frac{1}{2}x}{\frac{1}{2}x}\right)$$ is clearly an odd function of $x$. Also $$\frac{dy}{dx}=\frac{1}{x^2}\{\frac{1}{2}x\coth \frac{1}{2}x–1–\log \left(\frac{\sinh \frac{1}{2}x}{\frac{1}{2}x}\right)\}.$$ The function inside the large bracket tends to zero as $x\to 0$; and its derivative is $$\frac{1}{x}\left\{1–{\left(\frac{\frac{1}{2}x}{\sinh \frac{1}{2}x}\right)}^2\right\},$$ which has the sign of $x$. Hence $dy/dx>0$ for all values of $x$.]

 

38. Trace the curve $y=e^{1/x}\sqrt{x^2+2x}$, and show that the equation $$e^{1/x}\sqrt{x^2+2x}=\alpha$$ has no real roots if $\alpha$ is negative, one negative root if $$0<\alpha <a=e^{1/\sqrt{2}}\sqrt{2+2\sqrt{2}},$$ and two positive roots and one negative if $\alpha >a$.

 

39. Show that the equation $f_n(x)=1+x+{\displaystyle \frac{x^2}{2!}}+\cdots +{\displaystyle \frac{x^n}{n!}}=0$ has one real root if $n$ is odd and none if $n$ is even.

[Assume this proved for $n=1$, $2$, … $2k$. Then $f_{2k+1}(x)=0$ has at least one real root, since its degree is odd, and it cannot have more since, if it had, $f_{2k+1}^{\prime }(x)$ or $f_{2k}(x)$ would have to vanish once at least. Hence $f_{2k+1}(x)=0$ has just one root, and so $f_{2k+2}(x)=0$ cannot have more than two. If it has two, say $\alpha$ and $\beta$, then $f_{2k+2}^{\prime }(x)$ or $f_{2k+1}(x)$ must vanish once at least between $\alpha$ and $\beta$, say at $\gamma$. And $$f_{2k+2}(\gamma )=f_{2k+1}(\gamma )+\frac{{\gamma }^{2k+2}}{(2k+2)!}>0.$$ But $f_{2k+2}(x)$ is also positive when $x$ is large (positively or negatively), and a glance at a figure will show that these results are contradictory. Hence $f_{2k+2}(x)=0$ has no real roots.]

 

40. Prove that if $a$ and $b$ are positive and nearly equal then $$\log \frac{a}{b}=\frac{1}{2}(a–b)(\frac{1}{a}+\frac{1}{b}),$$ approximately, the error being about $\frac{1}{6}\{(a–b)/a{\}}^3$. [Use the logarithmic series. This formula is interesting historically as having been employed by Napier for the numerical calculation of logarithms.]

 

41. Prove by multiplication of series that if $-1<x<1$ then $$\begin{array}{rl}\frac{1}{2}\{\log (1+x){\}}^2& =\frac{1}{2}{x}^2–\frac{1}{3}(1+\frac{1}{2})x^3+\frac{1}{4}(1+\frac{1}{2}+\frac{1}{3})x^4–\dots ,\\ \frac{1}{2}(\arctan x{)}^2& =\frac{1}{2}{x}^2–\frac{1}{4}(1+\frac{1}{3})x^4+\frac{1}{6}(1+\frac{1}{3}+\frac{1}{5})x^6–\dots .\end{array}$$

 

42. Prove that $$(1+\alpha x{)}^{1/x}=e^{\alpha }\{1–\frac{1}{2}{a}^{2}x+\frac{1}{24}(8+3a)a^3{x}^2(1+{ϵ}_x)\},$$ where ${ϵ}_x\to 0$ with $x$.

 

43. The first $n+2$ terms in the expansion of $\log (1+x+{\displaystyle \frac{x^2}{2!}}+\cdots +{\displaystyle \frac{x^n}{n!}})$ in powers of $x$ are $$x–\frac{x^{n+1}}{n!}\{\frac{1}{n+1}–\frac{x}{1!(n+2)}+\frac{x^2}{2!(n+3)}–\cdots +(-1{)}^n\frac{x^n}{n!(2n+1)}\}.$$

 

44. Show that the expansion of $$\exp (-x–\frac{x^2}{2}–\cdots –\frac{x^n}{n})$$ in powers of $x$ begins with the terms $$1–x+\frac{x^{n+1}}{n+1}–\sum _{s=1}^n\frac{x^{n+s+1}}{(n+s)(n+s+1)}.$$

 

45. Show that if $-1<x<1$ then $$\begin{array}{rl}\frac{1}{3}x+\frac{1\cdot 4}{3\cdot 6}{2}^2{x}^2+\frac{1\cdot 4\cdot 7}{3\cdot 6\cdot 9}{3}^2{x}^3+\dots & =\frac{x(x+3)}{9(1–x{)}^{7/3}},\\ \frac{1}{3}x+\frac{1\cdot 4}{3\cdot 6}{2}^3{x}^2+\frac{1\cdot 4\cdot 7}{3\cdot 6\cdot 9}{3}^3{x}^3+\dots & =\frac{x(x^2+18x+9)}{27(1–x{)}^{10/3}}.\end{array}$$

[Use the method of Ex. XCII. 6. The results are more easily obtained by differentiation; but the problem of the differentiation of an infinite series is beyond our range.]

 

46. Prove that $$\begin{array}{rl}{\int }_0^{\mathrm{\infty }}\frac{dx}{(x+a)(x+b)}& =\frac{1}{a–b}\log \left(\frac{a}{b}\right),\\ {\int }_0^{\mathrm{\infty }}\frac{dx}{(x+a)(x+b{)}^2}& =\frac{1}{(a–b{)}^{2}b}\{a–b–b\log \left(\frac{a}{b}\right)\},\\ {\int }_0^{\mathrm{\infty }}\frac{xdx}{(x+a)(x+b{)}^2}& =\frac{1}{(a–b{)}^2}\{a\log \left(\frac{a}{b}\right)–a+b\},\\ {\int }_0^{\mathrm{\infty }}\frac{dx}{(x+a)(x^2+b^2)}& =\frac{1}{(a^2+b^2)b}\{\frac{1}{2}\pi a–b\log \left(\frac{a}{b}\right)\},\\ {\int }_0^{\mathrm{\infty }}\frac{xdx}{(x+a)(x^2+b^2)}& =\frac{1}{a^2+b^2}\{\frac{1}{2}\pi b+a\log \left(\frac{a}{b}\right)\},\end{array}$$ provided that $a$ and $b$ are positive. Deduce, and verify independently, that each of the functions $$a–1–\log a,a\log a–a+1,\frac{1}{2}\pi a–\log a,\frac{1}{2}\pi +a\log a$$ is positive for all positive values of $a$.

 

47. Prove that if $\alpha$, $\beta$, $\gamma$ are all positive, and ${\beta }^2>\alpha \gamma$, then $${\int }_0^{\mathrm{\infty }}\frac{dx}{\alpha x^2+2\beta x+\gamma }=\frac{1}{\sqrt{{\beta }^2–\alpha \gamma }}\log \left\{\frac{\beta +\sqrt{{\beta }^2–\alpha \gamma }}{\sqrt{\alpha \gamma }}\right\};$$ while if $\alpha$ is positive and $\alpha \gamma >{\beta }^2$ the value of the integral is $$\frac{1}{\sqrt{\alpha \gamma –{\beta }^2}}\arctan \left\{\frac{\sqrt{\alpha \gamma –{\beta }^2}}{\beta }\right\},$$ that value of the inverse tangent being chosen which lies between $0$ and $\pi$. Are there any other really different cases in which the integral is convergent?

 

48. Prove that if $a>-1$ then $${\int }_1^{\mathrm{\infty }}\frac{dx}{(x+a)\sqrt{x^2–1}}={\int }_0^{\mathrm{\infty }}\frac{dt}{\cosh t+a}=2{\int }_1^{\mathrm{\infty }}\frac{du}{u^2+2au+1};$$ and deduce that the value of the integral is $$\frac{2}{\sqrt{1–a^2}}\arctan \sqrt{\frac{1–a}{1+a}}$$ if $-1<a<1$, and $$\frac{1}{\sqrt{a^2–1}}\log \frac{\sqrt{a+1}+\sqrt{a–1}}{\sqrt{a+1}–\sqrt{a–1}}=\frac{2}{\sqrt{a^2–1}}\arg \tanh \sqrt{\frac{a–1}{a+1}}$$ if $a>1$. Discuss the case in which $a=1$.

 

49. Transform the integral ${\int }_0^{\mathrm{\infty }}\frac{dx}{(x+a)\sqrt{x^2+1}}$, where $a>0$, in the same ways, showing that its value is $$\frac{1}{\sqrt{a^2+1}}\log \frac{a+1+\sqrt{a^2+1}}{a+1–\sqrt{a^2+1}}=\frac{2}{\sqrt{a^2+1}}\arg \tanh \frac{\sqrt{a^2+1}}{a+1}.$$

 

50. Prove that $${\int }_0^1\arctan xdx=\frac{1}{4}\pi –\frac{1}{2}\log 2.$$

 

51. If $0<\alpha <1$, $0<\beta <1$, then $${\int }_{-1}^1\frac{dx}{\sqrt{(1–2\alpha x+{\alpha }^2)(1–2\beta x+{\beta }^2)}}=\frac{1}{\sqrt{\alpha \beta }}\log \frac{1+\sqrt{\alpha \beta }}{1–\sqrt{\alpha \beta }}.$$

 

52. Prove that if $a>b>0$ then $${\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{d\theta }{a\cosh \theta +b\sinh \theta }=\frac{\pi }{\sqrt{a^2–b^2}}.$$

 

53. Prove that $${\int }_0^1\frac{\log x}{1+x^2}dx=-{\int }_1^{\mathrm{\infty }}\frac{\log x}{1+x^2}dx,{\int }_0^{\mathrm{\infty }}\frac{\log x}{1+x^2}dx=0,$$ and deduce that if $a>0$ then $${\int }_0^{\mathrm{\infty }}\frac{\log x}{a^2+x^2}dx=\frac{\pi }{2a}\log a.$$

[Use the substitutions $x=1/t$ and $x=au$.]

54. Prove that $${\int }_0^{\mathrm{\infty }}\log (1+\frac{a^2}{x^2})dx=\pi a$$ if $a>0$. [Integrate by parts.]

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$\leftarrow$ 216. An alternative method of development of the theory of the exponential and logarithmic functions

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