問7
次の3つの積分を求めましょう．
(1)
\begin{align*}
\int e^x\text{sin}5xdx &= e^x\sin5x – \int e^x5\cos5x dx
&=e^x\sin5x -5(e^x\cos5x+5\int e^x\sin5xdx)
&= e^x\sin5x -5e^x\cos5x-25\int e^x\sin5xdx
26\int e^x\text{sin}5xdx =& e^x(\sin5x+5\cos5x)
\int e^x\text{sin}5xdx &= \frac{e^x(\sin5x+5\cos5x)}{26}+C\quad(C:\text{積分定数})
\end{align*}
(2)
\begin{align*}
\int\text{cos}2x\text{cos}3xdx &= \int \frac{1}{2}(\cos(-x)+\cos5x)dx
&= \frac{1}{2}\{\int\cos xdx + \int\cos 5x dx \}
&= \frac{1}{2}(\sin x + \frac{\sin 5x}{5})+C\quad(C:\text{積分定数})
\end{align*}
(3)
\begin{align*}
\int_0^2dx\int_x^2e^{-y^2}dy
\end{align*}
(1)
\begin{align*}
\int e^x\text{sin}5xdx &= e^x\sin5x – \int e^x5\cos5x dx
&=e^x\sin5x -5(e^x\cos5x+5\int e^x\sin5xdx)
&= e^x\sin5x -5e^x\cos5x-25\int e^x\sin5xdx
26\int e^x\text{sin}5xdx =& e^x(\sin5x+5\cos5x)
\int e^x\text{sin}5xdx &= \frac{e^x(\sin5x+5\cos5x)}{26}+C\quad(C:\text{積分定数})
\end{align*}
(2)
\begin{align*}
\int\text{cos}2x\text{cos}3xdx &= \int \frac{1}{2}(\cos(-x)+\cos5x)dx
&= \frac{1}{2}\{\int\cos xdx + \int\cos 5x dx \}
&= \frac{1}{2}(\sin x + \frac{\sin 5x}{5})+C\quad(C:\text{積分定数})
\end{align*}
(3)
\begin{align*}
\int_0^2dx\int_x^2e^{-y^2}dy
\end{align*}