(1) ∫0∞x2e−x2 dx=∫0∞x⋅(xe−x2) dx=[−12xe−x2]0∞+12∫0∞e−x2 dx=π4\int_0^\infty x^2 e^{-x^2}\,dx = \int_0^\infty x \cdot (xe^{-x^2})\,dx = \left[-\frac{1}{2}xe^{-x^2}\right]_0^\infty + \frac{1}{2}\int_0^\infty e^{-x^2}\,dx = \frac{\sqrt{\pi}}{4}∫0∞x2e−x2dx=∫0∞x⋅(xe−x2)dx=[−21xe−x2]0∞+21∫0∞e−x2dx=4π (2) ∫0∞e−xx dx\int_0^\infty \frac{e^{-x}}{\sqrt{x}}\,dx∫0∞xe−xdx x=t\sqrt{x} = tx=t とおくと 12x dx=dt\dfrac{1}{2\sqrt{x}}\,dx = dt2x1dx=dt dx=2t dtdx = 2t\,dtdx=2tdt x:0→∞x : 0 \to \inftyx:0→∞ t:0→∞t : 0 \to \inftyt:0→∞ これから =∫0∞1te−t2⋅2t dt=2∫0∞e−t2 dt=π= \int_0^\infty \frac{1}{t}e^{-t^2} \cdot 2t\,dt = 2\int_0^\infty e^{-t^2}\,dt = \sqrt{\pi}=∫0∞t1e−t2⋅2tdt=2∫0∞e−t2dt=π