題目 9. (10 points) Compute the integral ∫01∫x1cos(y2+1) dy dx.\int_{0}^{1}\int_{x}^{1}\cos(y^2+1)\,dy\,dx.∫01∫x1cos(y2+1)dydx. 解答 解法一 交換積分次序後， ∫01∫x1cos(y2+1) dy dx=∫01ycos(y2+1) dy=sin2−sin12.\int_0^1\int_x^1\cos(y^2+1)\,\mathrm{d}y\,\mathrm{d}x =\int_0^1y\cos(y^2+1)\,\mathrm{d}y =\boxed{\frac{\sin2-\sin1}{2}}.∫01∫x1cos(y2+1)dydx=∫01ycos(y2+1)dy=2sin2−sin1.
