題目 3. (10 points) Evaluate the indefinite integral ∫sin(x) dx.\int \sin(\sqrt{x})\,dx.∫sin(x)dx. 解答 解法一 令 u=xu=\sqrt{x}u=x，則 dx=2u du\mathrm{d}x=2u\,\mathrm{d}udx=2udu： ∫sin(x) dx=2∫usinu du=−2xcos(x)+2sin(x)+C.\int\sin(\sqrt{x})\,\mathrm{d}x =2\int u\sin u\,\mathrm{d}u =\boxed{-2\sqrt{x}\cos(\sqrt{x})+2\sin(\sqrt{x})+C}.∫sin(x)dx=2∫usinudu=−2xcos(x)+2sin(x)+C.
