題目 Problem Compute the following integrals. i) (10%)(10\%)(10%) ∫x3ex2 dx.\int x^3e^{x^2}\,dx.∫x3ex2dx. ii) (10%)(10\%)(10%) ∫0∞dxx2+2x+2.\int_0^\infty \frac{dx}{x^2+2x+2}.∫0∞x2+2x+2dx. 解答 解法一 思路 展開 (i) 令 u=x2u=x^2u=x2，把多出的 x2x^2x2 寫成 uuu；(ii) 配方後按瑕積分取上限極限。 答題過程 展開 (i) ∫x3ex2 dx=12∫ueu du=12ex2(x2−1)+C.\int x^3e^{x^2}\,\mathrm{d}x =\frac12\int ue^u\,\mathrm{d}u =\boxed{\frac12e^{x^2}(x^2-1)+C}.∫x3ex2dx=21∫ueudu=21ex2(x2−1)+C. (ii) ∫0∞dxx2+2x+2=limb→∞[arctan(x+1)]0b=π4.\int_0^\infty\frac{\mathrm{d}x}{x^2+2x+2} =\lim_{b\to\infty}\bigl[\arctan(x+1)\bigr]_0^b =\boxed{\frac\pi4}.∫0∞x2+2x+2dx=b→∞lim[arctan(x+1)]0b=4π.
