積分100本ノック（解答編）

$ \displaystyle \frac{1}{2}\tan^2x+\log{|\cos x|}+C$

$ \displaystyle 4\sqrt{2}$

$ \displaystyle \frac{\pi}{2}-1$

$ \displaystyle \frac{1}{4\log{2}}$

$ \displaystyle \frac{1}{2}\log{\frac{1-\cos x}{1+\cos x}}+C$

$ \displaystyle \log{(x+\sqrt{x^2+1})}+C$

$ \displaystyle \frac{2}{45}(3x-5)\sqrt{3x-5}(3x+\frac{10}{3})+C$

$ \displaystyle \frac{2}{9}e\sqrt{e}+\frac{4}{9}$

$ \displaystyle x(\log{x})^2-2(x\log{x}-x)+C$

$ \displaystyle 6+\frac{\pi}{8}$

$ \displaystyle \log{|1+\log{x|}}+C$

$ \displaystyle 2\log{|\sqrt{x}-1|}+C$

$ \displaystyle \frac{\pi}{12}+\frac{1}{2}-\frac{\sqrt{3}}{4}$

$ \displaystyle \log{|x+3+\sqrt{x^2+6x+13}|}+C $

$ \displaystyle x\tan x+\log{|\cos x|}+C $

$ \displaystyle \frac{1}{\log{2}}(x \log{x} - x) + C$

$ \displaystyle - \log{|x+1|} + \frac{1}{2} \log{(x^2+1)} + C$

$ \displaystyle 2 \log{|x|} - 2 \log{|x+1|} + \frac{1}{x+1} + C $

$ \displaystyle - \frac{1}{\tan x} + C $

$ \displaystyle \frac{\pi}{4} - \frac{1}{2} $

$ \displaystyle -(\log{| \cos x |})^2+C$

$ \displaystyle \frac{1}{2} \frac{\sin \theta}{\cos \theta} + \frac{1}{4} \log{\left|\frac{1+\sin \theta}{1-\sin \theta}\right|}+C$

$ \displaystyle \tan x - \frac{1}{\cos x} + C$

$ \displaystyle \frac{x 2^x}{\log 2} - \frac{2^x}{(\log 2)^2} + C $

$ \displaystyle \frac{e^3}{3} - \frac{e^2}{2}$

$ \displaystyle \log{|\tan x|} +C$

$ \displaystyle \frac{x 2^{\log x}}{\log 2 + 1} + C$

$ \displaystyle \frac{1}{3}\log 2 + \frac{\sqrt{3}}{9}\pi$

$ \displaystyle 1 $

$ \displaystyle \tan x + \frac{1}{3} \tan^3 x + C$

$ \displaystyle \frac{1}{3} $

$ \displaystyle \frac{\cos \frac{1}{x}}{x} - \sin \frac{1}{x} +C$

$ \displaystyle \sqrt{\tan x} + C$

$ \displaystyle \frac{1}{2}xe^x(\sin x - \cos x) + \frac{1}{2}e^x \cos x + C $

$ \displaystyle \frac{1}{2} x \{ \sin(\log x) - \cos(\log x) \} + C$

$ \displaystyle \frac{1}{10}(9\sqrt{3} -1)$

$ \displaystyle \frac{1}{2} \{\log(\log x)\}^2 + C$

$ \displaystyle \frac{\pi}{4}$

$ \displaystyle 2 e^\frac{x}{2} + C $

$ \displaystyle -\frac{1}{\log x}+ C$

$ \displaystyle -\frac{1}{4}x \cos{2x} + \frac{1}{8} \sin{2x} + C $

$ \displaystyle \frac{1}{50}e^x(5x+3) \sin2x + \frac{1}{25}e^x(2-5x) \cos2x + C$

$ \displaystyle \frac{\pi}{6}\log2 $

$ \displaystyle \frac{\pi}{28} $

$ \displaystyle -\frac{1}{3} \log{|1+x^{-3}|} + C$

$ \displaystyle 4040$

$ \displaystyle e^3 - e^2$

$ \displaystyle x^x + C$

$ \displaystyle \frac{1263}{56}$

$ \displaystyle \log{|e^x -1|}+ C$

$ \displaystyle \frac{\pi}{12} $

$ \displaystyle \log2 - 2 + \frac{\pi}{2}$

$ \displaystyle \frac{191}{12} $

$ \displaystyle \frac{\pi^2}{8} $

$ \displaystyle -\frac{\log x + 1}{x} + C$

$ \displaystyle -\frac{(\log x)^2 + 2 \log x + 2}{x} + C$

$ \displaystyle \frac{1}{12} \sin 6x + \frac{1}{4} \sin 2x + C $

$ \displaystyle \frac{1}{4}(- \log{|2-\log x|} + \log{|2+\log x| }) + C$

$ \displaystyle \frac{\pi}{2}$

$ \displaystyle \frac{\pi}{12} $

$ \displaystyle \frac{1}{2} \log{|\sqrt{1-x^2} -1|} - \frac{1}{2} \log{|\sqrt{1-x^2} +1|} + C $

$ \displaystyle e^{e^x} + C$

$ \displaystyle \frac{3^{-2+5x}}{5\log 3} +C$

$ \displaystyle \frac{1}{3}(\log x +3)^3 + C $

$ \displaystyle \left( \frac{1}{2} - \frac{1}{\sqrt{3}} \right)\log2 + \frac{\pi}{12}$

$ \displaystyle \frac{1}{2}\log{\frac{x^2+2}{x^2+3}} + C$

$ \displaystyle \frac{\pi}{4} $

$ \displaystyle 1$

$ \displaystyle \log 3 + \frac{\sqrt{3}}{9}\pi $

$ \displaystyle \frac{4-\pi}{4+ \pi} $

$ \displaystyle 0 $

$ \displaystyle \frac{1}{2}\log{\frac{|x(x+2)|}{(x+1)^2}} + C$

$ \displaystyle \log{\left|\frac{1}{(x-1)^3(x+1)}\right|} - \frac{10}{x-1} +C$

$ \displaystyle -\frac{1}{2}\log{|1-x^2|} + C $

$ \displaystyle \frac{1}{2}$

$ \displaystyle \frac{1}{4}\{ (2x^2-4x-1) \sin{2x} +2(x-1) \cos{2x} \} + C$

$ \displaystyle -\frac{65}{e} + 24 $

$ \displaystyle \frac{74}{9} $

$ \displaystyle \frac{1}{2}\log{\left(\frac{1+\sin x}{1-\sin x}\right)} +C$

$ \displaystyle \frac{1}{5}(e^{2 \pi} + 1) $

$ \displaystyle \frac{1}{5} e^x (\sin^2 x - \sin{2x} + C) $

$ \displaystyle \frac{1}{3}(1-\log 2)$

$ \displaystyle \sqrt{2} - \log(\sqrt{2}+1) $

$ \displaystyle \frac{\pi}{16} $

$ \displaystyle \frac{1}{\log 3} \log (3^x + \log 3) +C $

$ \displaystyle \frac{1}{4} \tan^4 x - \frac{1}{2}\tan^2 x - \log{|\cos x|} + C $

$ \displaystyle 1$

$ \displaystyle \frac{\pi}{8} \log 2 $

$ \displaystyle \frac{1}{48}(12x +6 \sin{2x} + 3 \sin{4x} + 2 \sin{6x}) +C$

$ \displaystyle \frac{\pi}{4} - \frac{1}{2} \log 2 $

$ \displaystyle \frac{\pi}{2} $

$ \displaystyle \frac{1}{2}x^2 + C$

$ \displaystyle \frac{1}{2} \left\{ \frac{x(x+2)}{(x+1)^2}\log x - \log(x+1) + \frac{1}{x+1} \right\} + C $

$ \displaystyle \frac{e^2 + 1}{4} $

$ \displaystyle \log(1-x^2) + x\log{\frac{1+x}{1-x}} +C $

$ \displaystyle 0$

$ \displaystyle x^{2x} +C$

$ \displaystyle 2(\sin 1 + \cos 1 - 1) $

$ \displaystyle 4 \sqrt{10} $

$ \displaystyle \frac{1}{\sqrt{2}} \left\{ \log(\sqrt{2}-1) + \frac{\pi}{2} \right\} $