参考にしたサイト➔部分積分の公式と覚え方,例題 | 高校数学の美しい物語
解説:
\( \hspace{35px} \begin{eqnarray} \displaystyle \int_{}^{} \log x dx & = & x\log x - \displaystyle \int_{}^{} 1 dx\\ & = & x\log x - x + C (Cは積分定数) \\ \end{eqnarray} \)解説:
\( \hspace{35px} \begin{eqnarray} \displaystyle \int_{}^{} x^2\sin x dx & = & -x^2\cos x + 2x\sin x + 2\cos x + C (Cは積分定数) \\ \end{eqnarray} \)解説:
\( \displaystyle \int_{}^{} \sin x\cos x dx = I \) とすると解説:
\( \hspace{35px} \begin{eqnarray} \displaystyle \int_{1}^{6} x\log x dx & = & \left[ \dfrac{x^2}{2}\log x \right]_{1}^{6} - \displaystyle \int_{1}^{6} \dfrac{x}{2} dx \\ & = & \left[ \dfrac{x^2}{2}\log x \right]_{1}^{6} - \left[ \dfrac{x^2}{4} \right]_{1}^{6} \\ & = & \left[ \dfrac{x^2}{2}\log x - \dfrac{x^2}{4} \right]_{1}^{6} \\ & = & 18\log 6 - 9 - (-\dfrac{1}{4}) \\ & = & 18\log 6 - \dfrac{35}{4} \end{eqnarray} \)解説:
\( \hspace{35px} \begin{eqnarray} \displaystyle \int_{0}^{4} (2 - x)^2 e^x dx & = & \left[ (x^2 - 6x + 10)e^x \right]_{0}^{4} \\ & = & 2e^4 - 10 \\ \end{eqnarray} \)