題目 4. (10 points) Evaluate the following integral: ∫01∫x1xcos(y5+2) dy dx.\int_{0}^{1}\int_{\sqrt{x}}^{1} x\cos(y^5+2)\,dy\,dx.∫01∫x1xcos(y5+2)dydx. 解答 解法一 交換積分次序後 0≤y≤10\le y\le10≤y≤1、0≤x≤y20\le x\le y^20≤x≤y2，故 ∫01∫0y2xcos(y5+2) dx dy=12∫01y4cos(y5+2) dy=sin3−sin210.\int_0^1\int_0^{y^2}x\cos(y^5+2)\,\mathrm{d}x\,\mathrm{d}y =\frac12\int_0^1y^4\cos(y^5+2)\,\mathrm{d}y =\boxed{\frac{\sin3-\sin2}{10}}.∫01∫0y2xcos(y5+2)dxdy=21∫01y4cos(y5+2)dy=10sin3−sin2.
