参考にしたサイト➔積分公式一覧 | 高校数学の美しい物語
解説:
\( \hspace{35px} \begin{eqnarray} \displaystyle \int_{0}^{1} x^{2024} dx & = & \left[ \dfrac{1}{2025}x^{2025} \right]_{0}^{1}\\ & = & \dfrac{1}{2025} \\ \end{eqnarray} \)解説:
\( \hspace{35px} \begin{eqnarray} \displaystyle \int_{100}^{200} e^{x - 100} dx & = & e^{-100}\displaystyle \int_{100}^{200} e^x dx \\ & = & e^{-100}\left[ e^x \right]_{100}^{200} & = & e^{100} - 1 \end{eqnarray} \)解説:
\( \hspace{35px} \begin{eqnarray} \displaystyle \int \dfrac{1}{x\log 15} dx & = & \dfrac{1}{\log 15}\displaystyle \int \dfrac{1}{x} dx \\ & = & \dfrac{1}{\log 15}\log|x| + C \\ & = & \log_{15} e \log_e |x| + C \\ & = & \log_{15} |x| + C (Cは積分定数) \end{eqnarray} \)解説:
\( \hspace{35px} \begin{eqnarray} \displaystyle \int \biggl( \dfrac{\sqrt{x}}{2x} - \dfrac{2}{x^3} \biggr) dx & = & \displaystyle \int \biggl( \dfrac{1}{2\sqrt{x}} - \dfrac{2}{x^3} \biggr) dx \\ & = & \displaystyle \int \biggl( \dfrac{1}{2}x^{-\frac{1}{2}} - 2x^{-3} \biggr) dx \\ & = & x^{\frac{1}{2}} + x^{-2} + C \\ & = & \sqrt{x} + \dfrac{1}{x^2} + C (Cは積分定数) \end{eqnarray} \)解説:
\( \hspace{35px} \begin{eqnarray} \displaystyle \int \sin^2 \dfrac{x}{2} dx & = & \displaystyle \int \biggl( \dfrac{1 - \cos x}{2} \biggr) dx \\ & = & \dfrac{1}{2}\displaystyle \int (1 - \cos x) dx \\ & = & \dfrac{1}{2}(x - \sin x) + C (Cは積分定数) \end{eqnarray} \)