{"id":88982,"date":"2026-08-02T08:17:04","date_gmt":"2026-08-02T00:17:04","guid":{"rendered":"https:\/\/www.wsisp.com\/helps\/88982.html"},"modified":"2026-08-02T08:17:04","modified_gmt":"2026-08-02T00:17:04","slug":"%e9%ab%98%e7%ad%89%e6%95%b0%e5%ad%a6%e5%ad%a6%e4%b9%a0%e6%95%99%e7%a8%8b%ef%bc%8c%e4%bb%8e%e5%85%a5%e9%97%a8%e5%88%b0%e7%b2%be%e9%80%9a%ef%bc%8c%e5%ae%9a%e7%a7%af%e5%88%86-%e7%9f%a5%e8%af%86","status":"publish","type":"post","link":"https:\/\/www.wsisp.com\/helps\/88982.html","title":{"rendered":"\u9ad8\u7b49\u6570\u5b66\u5b66\u4e60\u6559\u7a0b\uff0c\u4ece\u5165\u95e8\u5230\u7cbe\u901a\uff0c\u5b9a\u79ef\u5206 \u2014 \u77e5\u8bc6\u70b9\u5168\u9762\u603b\u7ed3(10\uff09"},"content":{"rendered":"

\u7b2c\u56db\u8282 \u53cd\u5e38\u79ef\u5206<\/h3>\n

\u4e00\u3001\u65e0\u7a77\u9650\u7684\u53cd\u5e38\u79ef\u5206<\/h4>\n

\u5b9a\u4e491： \u8bbe f(x)f(x)<\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span><\/span><\/span><\/span> \u5728 [a,+\u221e)[a, +\\\\infty)<\/span><\/span>[<\/span>a<\/span>,<\/span><\/span>+<\/span>\u221e<\/span>)<\/span><\/span><\/span><\/span><\/span> \u4e0a\u8fde\u7eed，\u82e5\u6781\u9650 lim\u2061b\u2192+\u221e\u222babf(x)\u2009dx\\\\lim_{b \\\\to +\\\\infty}\\\\int_a^b f(x)\\\\,dx<\/span><\/span>lim<\/span><\/span>b<\/span>\u2192<\/span>+<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>a<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>d<\/span>x<\/span><\/span><\/span><\/span><\/span> \u5b58\u5728，\u5219\u79f0\u53cd\u5e38\u79ef\u5206 \u222ba+\u221ef(x)\u2009dx\\\\int_a^{+\\\\infty} f(x)\\\\,dx<\/span><\/span>\u222b<\/span><\/span>a<\/span><\/span><\/span><\/span>+<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>d<\/span>x<\/span><\/span><\/span><\/span><\/span> \u6536\u655b，\u5e76\u5b9a\u4e49：<\/p>\n

\u222ba+\u221ef(x)\u2009dx=lim\u2061b\u2192+\u221e\u222babf(x)\u2009dx\\\\int_a^{+\\\\infty} f(x)\\\\,dx = \\\\lim_{b \\\\to +\\\\infty}\\\\int_a^b f(x)\\\\,dx<\/span><\/span>\u222b<\/span><\/span>a<\/span><\/span><\/span><\/span>+<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>d<\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>b<\/span>\u2192<\/span>+<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>a<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>d<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u82e5\u6781\u9650\u4e0d\u5b58\u5728，\u5219\u79f0\u8be5\u53cd\u5e38\u79ef\u5206\u53d1\u6563\u3002<\/p>\n

\u7c7b\u4f3c\u5b9a\u4e49：<\/p>\n

\u222b\u2212\u221ebf(x)\u2009dx=lim\u2061a\u2192\u2212\u221e\u222babf(x)\u2009dx\\\\int_{-\\\\infty}^b f(x)\\\\,dx = \\\\lim_{a \\\\to -\\\\infty}\\\\int_a^b f(x)\\\\,dx<\/span><\/span>\u222b<\/span><\/span>\u2212<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>d<\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>a<\/span>\u2192<\/span>\u2212<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>a<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>d<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u222b\u2212\u221e+\u221ef(x)\u2009dx=\u222b\u2212\u221ecf(x)\u2009dx+\u222bc+\u221ef(x)\u2009dx\\\\int_{-\\\\infty}^{+\\\\infty} f(x)\\\\,dx = \\\\int_{-\\\\infty}^{c} f(x)\\\\,dx + \\\\int_c^{+\\\\infty} f(x)\\\\,dx<\/span><\/span>\u222b<\/span><\/span>\u2212<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>+<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>d<\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span>\u222b<\/span><\/span>\u2212<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>c<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>d<\/span>x<\/span><\/span>+<\/span><\/span><\/span><\/span>\u222b<\/span><\/span>c<\/span><\/span><\/span><\/span>+<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>f<\/span>(<\/span>x<\/span>)<\/span><\/span>d<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n

\u5176\u4e2d cc<\/span><\/span>c<\/span><\/span><\/span><\/span><\/span> \u4e3a\u4efb\u610f\u5b9e\u6570\u3002\u53f3\u7aef\u4e24\u4e2a\u79ef\u5206\u90fd\u6536\u655b\u65f6，\u5de6\u7aef\u624d\u6536\u655b\u3002<\/p>\n

\u5178\u578b\u4f8b\u9898：<\/p>\n

\u4f8b5： \u8ba8\u8bba \u222b1+\u221e1xp\u2009dx\\\\displaystyle\\\\int_1^{+\\\\infty} \\\\frac{1}{x^p}\\\\,dx<\/span><\/span>\u222b<\/span><\/span>1<\/span><\/span><\/span><\/span>+<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>x<\/span><\/span><\/span><\/span><\/span>（pp<\/span><\/span>p<\/span><\/span><\/span><\/span><\/span> \u4e3a\u5e38\u6570）\u3002<\/p>\n

\u89e3： \u5f53 p\u22601p \\\\neq 1<\/span><\/span>p<\/span><\/span><\/span>\ue020<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span><\/span> \u65f6：<\/p>\n

\u222b1+\u221e1xp\u2009dx=lim\u2061b\u2192+\u221e\u222b1bx\u2212p\u2009dx=lim\u2061b\u2192+\u221e[x1\u2212p1\u2212p]1b=lim\u2061b\u2192+\u221eb1\u2212p\u221211\u2212p\\\\int_1^{+\\\\infty}\\\\frac{1}{x^p}\\\\,dx = \\\\lim_{b\\\\to+\\\\infty}\\\\int_1^b x^{-p}\\\\,dx = \\\\lim_{b\\\\to+\\\\infty}\\\\left[\\\\frac{x^{1-p}}{1-p}\\\\right]_1^b = \\\\lim_{b\\\\to+\\\\infty}\\\\frac{b^{1-p}-1}{1-p}<\/span><\/span>\u222b<\/span><\/span>1<\/span><\/span><\/span><\/span>+<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>b<\/span>\u2192<\/span>+<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u222b<\/span><\/span>1<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>\u2212<\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>d<\/span>x<\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>b<\/span>\u2192<\/span>+<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>[<\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span>x<\/span><\/span>1<\/span>\u2212<\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>]<\/span><\/span><\/span><\/span>1<\/span><\/span><\/span><\/span>b<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>=<\/span><\/span><\/span><\/span><\/span>b<\/span>\u2192<\/span>+<\/span>\u221e<\/span><\/span><\/span><\/span><\/span>lim<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>1<\/span><\/span>\u2212<\/span><\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span>b<\/span><\/span>1<\/span>\u2212<\/span>p<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212<\/span><\/span>1<\/span><\/span><\/span><\/span>\u200b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/p>\n