LaTeX 解答生成・検査パイプライン

実行日時: 2026-08-12T13:27:25（ID: 20260812-132725-7cdb22） ← 履歴一覧に戻る

¥item 次の不定積分を求めよ。
  ¥begin{edaenumerate}
  ¥item $¥displaystyle ¥int_{0}^{¥sqrt{¥pi}} 2x ¥cos x^2 dx$ ¥vspace{25ZW}
  ¥item $¥displaystyle ¥int_{0}^{1} x e^x dx$ ¥vspace{25ZW}
  ¥item $¥displaystyle ¥int_{2}^{3} ¥dfrac{1}{(x-1)(x+1)} dx$ ¥vspace{25ZW}
  ¥item $¥displaystyle ¥int_{1}^{2} ¥dfrac{2x+1}{x^2+x-1} dx$
  ¥item $¥displaystyle ¥int_{0}^{¥frac{¥pi}{2}} x ¥cos x dx$
 ¥item $¥displaystyle ¥int_{0}^{¥frac{¥pi}{2}} ¥sqrt{1-¥cos x} d x$ ¥vspace{22ZW}
 ¥item $¥displaystyle ¥int_{-2}^{2} ¥frac{(x+1)^{2}}{x^{2}+1} d x$ ¥vspace{22ZW}
 ¥item $¥displaystyle ¥int_{1}^{3} ¥frac{1}{x^{2}+3 x} d x$ ¥vspace{22ZW}
 ¥item $¥displaystyle ¥int_{e}^{e^{3}} x^{2} ¥log x d x$ ¥vspace{22ZW}
 ¥item $¥displaystyle ¥int_{0}^{¥frac{¥pi}{2}} ¥frac{¥sin x}{(1+¥cos x)^{3}} d x$
 ¥item $¥displaystyle ¥int_{0}^{¥frac{¥pi}{3}} ¥frac{1}{¥cos x} d x$ ¥vspace{23ZW}
¥item $¥displaystyle ¥int_{0}^{1} ¥frac{1}{¥sqrt{4-x^{2}}} d x$
¥item $¥displaystyle ¥int_{0}^{1} e^{x}¥left(e^{x}+1¥right)^{4} d x$ ¥vspace{23ZW}
¥item $¥displaystyle ¥int_{0}^{¥frac{¥pi}{4}} ¥tan ^{4} x¥left(1+¥tan ^{2} x¥right) d x$
¥item $¥displaystyle ¥int_{0}^{1} ¥frac{3 x-2}{e^{x}} d x$ ¥vspace{23ZW}
¥item $¥displaystyle ¥int_{-1}^{¥sqrt{3}-2} ¥frac{1}{x^{2}+4 x+5} d x$
¥item $¥displaystyle ¥int_{6}^{8} ¥frac{x}{x^{2}-6 x+8} d x$¥vspace{23ZW}
¥item $¥displaystyle ¥int_{0}^{¥frac{¥pi}{2}} ¥frac{¥sin 2 x}{3+¥cos ^{2} x} d x$
¥item $¥displaystyle ¥int_{0}^{¥frac{¥pi}{2}} x^{2} ¥cos ^{2} x d x$ ¥vspace{23ZW}
¥item $¥displaystyle ¥int_{1}^{e} ¥frac{¥sin (¥pi ¥log x)}{x} d x$
¥item $¥displaystyle ¥int_{0}^{1} ¥sqrt{2 x-x^{2}} d x$ ¥vspace{22ZW}
¥item $¥displaystyle ¥int_{0}^{¥frac{¥pi^{2}}{4}} ¥sin ¥sqrt{x} dx$
¥item $¥displaystyle ¥int_{0}^{¥frac{¥pi}{2}} x^3 ¥sin x d x$ ¥vspace{22ZW}
¥item $¥displaystyle ¥int_{0}^{1} x^{2} e^{2 x} d x$
¥item $¥displaystyle ¥int_{0}^{¥frac{¥pi}{3}} ¥frac{x}{¥cos ^{2} x} d x$ ¥vspace{23ZW}
¥item $¥displaystyle ¥int_{0}^{¥frac{¥pi}{2}} e^{x} ¥sin x d x$
  ¥end{edaenumerate}

\item
\begin{edaenumerate}
  \item $0$
  \item $1$
  \item $\dfrac{1}{2}\log\dfrac{3}{2}$
  \item $\log 5$
  \item $\dfrac{\pi}{2}-1$
  \item $2\sqrt{2}-2$
  \item $4$
  \item $\dfrac{1}{3}\log 2$
  \item $\dfrac{8e^9-2e^3}{9}$
  \item $\dfrac{3}{8}$
  \item $\log(2+\sqrt{3})$
  \item $\dfrac{\pi}{6}$
  \item $\dfrac{(e+1)^5-32}{5}$
  \item $\dfrac{1}{5}$
  \item $1-\dfrac{4}{e}$
  \item $\dfrac{\pi}{12}$
  \item $\log\dfrac{9}{8}$
  \item $\log\dfrac{4}{3}$
  \item $\dfrac{\pi(\pi^2-6)}{48}$
  \item $\dfrac{2}{\pi}$
  \item $\dfrac{\pi}{4}$
  \item $2$
  \item $\dfrac{3\pi^2}{4}-6$
  \item $\dfrac{e^2-1}{4}$
  \item $\dfrac{\pi}{\sqrt{3}}-\log 2$
  \item $\dfrac{e^{\pi/2}+1}{2}$
\end{edaenumerate}

- 17番目の答え($\displaystyle \int_{6}^{8} \frac{x}{x^{2}-6 x+8} d x$)が誤っています。

  部分分数分解: $\dfrac{x}{(x-2)(x-4)}=\dfrac{-1}{x-2}+\dfrac{2}{x-4}$

  積分: $[-\log|x-2|+2\log|x-4|]_6^8$
  - $x=8$: $-\log6+2\log4=-\log6+\log16$
  - $x=6$: $-\log4+2\log2=-\log4+\log4=0$

  よって値は $\log16-\log6=\log\dfrac{16}{6}=\log\dfrac{8}{3}$ となり、解答の $\log\dfrac{9}{8}$ は誤りです。

\item
\begin{edaenumerate}
  \item $0$
  \item $1$
  \item $\dfrac{1}{2}\log\dfrac{3}{2}$
  \item $\log 5$
  \item $\dfrac{\pi}{2}-1$
  \item $2\sqrt{2}-2$
  \item $4$
  \item $\dfrac{1}{3}\log 2$
  \item $\dfrac{8e^9-2e^3}{9}$
  \item $\dfrac{3}{8}$
  \item $\log(2+\sqrt{3})$
  \item $\dfrac{\pi}{6}$
  \item $\dfrac{(e+1)^5-32}{5}$
  \item $\dfrac{1}{5}$
  \item $1-\dfrac{4}{e}$
  \item $\dfrac{\pi}{12}$
  \item $\log\dfrac{8}{3}$
  \item $\log\dfrac{4}{3}$
  \item $\dfrac{\pi(\pi^2-6)}{48}$
  \item $\dfrac{2}{\pi}$
  \item $\dfrac{\pi}{4}$
  \item $2$
  \item $\dfrac{3\pi^2}{4}-6$
  \item $\dfrac{e^2-1}{4}$
  \item $\dfrac{\pi}{\sqrt{3}}-\log 2$
  \item $\dfrac{e^{\pi/2}+1}{2}$
\end{edaenumerate}