自作問題まとめ

はじめに

過去に作問した問題の一覧となります。積分や級数を解きたくなった時にぜひ活用ください。
横に解説と書いてある問題は既に解説記事を書いたものです。クリックすれば解説ページに飛びます。
注意ですが、問題は難易度順には並んでいません。

級数

1.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{H_n{H}_{n-1}}{n^2}}$ 解説

2.${\displaystyle \sum _{0<a<b}\frac{(-1{)}^b}{a^{2}b}}$ 解説

3.${\displaystyle \sum _{0<a<b}\frac{1}{a{b}^2{2}^b}}$ 解説

4.${\displaystyle \sum _{0\le n,k}\frac{(-1{)}^{n+k}}{(2n+1)(2k+1)(2n+2k+1)}}$ 解説

5.${\displaystyle \sum _{n=3}^{\mathrm{\infty }}\frac{(-1{)}^n{F}_{2n}}{F_n^4-1}}$ 解説

6.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{(4n)!}{(n-1)!(n+2)!(2n)!2^{6n}}}$

7.${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{(-1{)}^n}{1+{\pi }^2{n}^2}}$

8.${\displaystyle \sum _{n=2}^{\mathrm{\infty }}\frac{(-1{)}^n}{n^2-1}}$

9.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{\text{Ci}(\pi n)}{{\pi }^2{n}^2}}$

10.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{2n}{n})\frac{1}{n(n+1)2^{2n}}}$

11.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{\text{Si}({\pi }^3{n}^3)}{{\pi }^3{n}^3}}$

12.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{H_n^2}{n^3}}$

13.${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{n^3}{(\genfrac{}{}{0ex}{}{2n}{n})}}$

14.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}(\frac{1}{(2n-1{)}^2}-\frac{1}{(2n{)}^2})\psi (n)}$

15.${\displaystyle \sum _{0<n<k}\frac{1}{(2n-1{)}^2(2k-1{)}^2}}$

16.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{\text{Li}}_3(-e^{-(2n+1)\pi })}$

17.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{{\psi }^{\prime }(n)}{n^2(\genfrac{}{}{0ex}{}{2n}{n})}}$

18.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}(\zeta (6n)-1)}$

19.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n\zeta (2n)}{2^{2n}}}$

20.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}(2-(2n+1)\log (1+\frac{1}{n}))}$

21.${\displaystyle \sum _{p:prime}\frac{\log p}{p^2-1}}$

22.${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{3n^2-1}{(2n+1)(\genfrac{}{}{0ex}{}{2n}{n})}}$

23.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{2F_{n+1}^2-F_n^2}{F_n{F}_{n+1}{F}_{n+2}{F}_{n+3}}}$

24.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}\frac{(-1{)}^n\eta (n)}{n}}$

25.${\displaystyle \sum _{n=1}^{\mathrm{\infty }}{\psi }^{(2)}\left(\frac{n}{2}\right)}$

26.${\displaystyle \sum _{n=0}^{\mathrm{\infty }}\frac{(4n)!H_{n+1}}{n!(n+1)!(2n)!2^{6n}}}$

27.${\displaystyle \sum _{n=0}^{\mathrm{\infty }}{(\genfrac{}{}{0ex}{}{2n}{n})}^2\frac{H_n}{(n+1)2^{4n}}}$

28.${\displaystyle \sum _{n=0}^{\mathrm{\infty }}(\genfrac{}{}{0ex}{}{2n}{n})\frac{1}{(2n+1)2^{4n}}}$

29.${\displaystyle \sum _{0<n}\frac{1}{n}\sum _{\begin{array}{c}0<k\\ n\ne k\end{array}}\frac{1}{n^2-k^2}}$

積分

1.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{\sin x}{\sinh x}dx}$ 解説

2.${\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{\cos x\sqrt{\log \tan x}}{{\sin }^{3}x}dx}$ 解説

3.${\displaystyle {\int }_0^{\mathrm{\infty }}\log \left(\frac{e^x}{e^x-1}\right)dx}$ 解説

4.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{1}{\sqrt{x}(x+1)}dx}$ 解説

5.${\displaystyle {\int }_0^1\frac{\log \left(\frac{1+x}{1-x}\right)}{x}dx}$ 解説

6.${\displaystyle {\int }_0^{\frac{\pi }{2}}{e}^{-\frac{\pi }{2}\tan x}dx}$ 解説

7.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{e^{-x}{\sin }^{2}x}{x}dx}$ 解説

8.${\displaystyle {\int }_0^{\frac{\pi }{2}}\cos \log \tan xdx}$ 解説

9.${\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{\arctan \frac{\tan x}{\cos x}}{\cos x}}$ 解説

10.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{\sin \pi x}{x{e}^x}dx}$ 解説

11.${\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{1}{1+{\cos }^{4}x}dx}$ 解説

12.${\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{\log \cos x}{\cos x}dx}$ 解説

13.${\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{{\arctan }^{2}x}{x^2}dx}$ 解説

14.${\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{\arctan x^2}{x^2}dx}$ 解説

15.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{e^{\frac{1}{6}x}+e^{\frac{5}{6}x}-2e^{\frac{1}{2}x}}{x(e^x-1)}dx}$

16.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{x^2{e}^{\frac{1}{2}x}}{e^x+1}dx}$

17.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{1-\tanh x}{\sqrt{\tanh x}}dx}$

18.${\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{\sin \log \tan x}{\log \tan x}dx}$

19.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{1}{\sqrt{x}(x+1)}dx}$

20.${\displaystyle {\int }_0^1\frac{\log x}{\sqrt{x}(x+1)}dx}$

21.${\displaystyle {\int }_0^{\mathrm{\infty }}\log (1-e^{-x})dx}$

22.${\displaystyle {\int }_0^1\frac{\log x}{\sqrt{x}-x}dx}$

23.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{\log (x+1)}{x\sqrt[4]{x}}dx}$

24.${\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{\cos \log \tan x\log \cos x}{{\sin }^{2}x}dx}$

25.${\displaystyle {\int }_0^{1}x\log \log \frac{1}{x}dx}$

26.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{\log x\log (1+x)}{x(1+x)}}$

27.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{\cos x\arctan x}{x}dx}$

28.${\displaystyle {\int }_{-1}^0\frac{{\text{Li}}_2(x)}{1-x}}$

29.${\displaystyle {\int }_0^1\frac{\log x\log (1+x)}{x(1-x)}dx}$

30.${\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{\log \cos x}{\tan x}dx}$

31.${\displaystyle {\int }_0^1\frac{{\text{Li}}_2(x)\log x}{x(1+x)}dx}$

32.${\displaystyle {\int }_0^1\frac{{\log }^{3}x{\text{Li}}_4(x)}{1-x}dx}$

33.${\displaystyle {\int }_0^{\frac{1}{2}}\frac{{\text{arsinh}}^{2}x}{x}dx}$

34.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{I_0(\sqrt{x})}{e^x}dx}$

35.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{x\log \tanh x}{\tanh x}dx}$

36.${\displaystyle {\int }_0^{\mathrm{\infty }}x\log \tanh xdx}$

37.${\displaystyle {\int }_0^{\frac{\pi }{2}}x\log 2\sin xdx}$

38.${\displaystyle {\int }_0^{\frac{\pi }{2}}{x}^2\log 2\sin xdx}$

39.${\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{\sin \frac{x}{2}}{\sqrt{\cos x}\text{artanh}\sin x}dx}$

40.${\displaystyle {\int }_0^{\mathrm{\infty }}{e}^{-x}{\log }^{2}xdx}$

41.${\displaystyle {\int }_0^1(\frac{x}{\log x}+\frac{1}{1-x})dx}$

42.${\displaystyle {\int }_0^1\frac{\gamma (x){\log }^{2}x}{x}dx}$

43.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{\sin x}{1+e^{\pi x}}dx}$

44.${\displaystyle {\int }_0^{\frac{\pi }{2}}\tan x\log \sin x\log \cos xdx}$

45.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{{\sin }^{3}x}{x(1+{\sin }^{4}x)}dx}$

46.${\displaystyle {\int }_0^1\text{li}(x)\cos \log xdx}$

47.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{\sin x{\log }^2\sin x}{{\cos }^{3}x}dx}$

48.${\displaystyle {\int }_0^{\frac{\pi }{2}}\frac{1+2\cos 2x}{3+\cos 4x}dx}$

49.${\displaystyle {\int }_0^{\mathrm{\infty }}\frac{(x-1{)}^2}{(x+1)(x^3-1)}dx}$

50.${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{\text{arcoth}\sqrt{x^2+{\pi }^2}}{\sqrt{x^2+{\pi }^2}}dx}$

51.${\displaystyle {\int }_0^{\mathrm{\infty }}(\frac{\text{artanhsech}x}{\text{sech}x}-1)dx}$

52.${\displaystyle {\int }_0^{\frac{\pi }{2}}\log (\frac{9}{16}+x^2)dx}$

53.${\displaystyle {\int }_0^1(\frac{1}{2(1-x)}-\frac{1}{(1-x)\log x}-\frac{1}{{\log }^{2}x})dx}$

最後に

今後まだ解説を書いていないものの解説を書いていくつもりです。