\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n
(1)<\/strong>\u3000\u5b9a\u7a4d\u5206\r\n\\[\r\n\\displaystyle\\int _ {1}^{e} x^{\\frac{1}{n}} \\log x \\, dx \\quad ( n =1, 2, 3, \\cdots )\r\n\\]\r\n\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\u6975\u9650\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} n \\left( \\displaystyle\\int _ {1}^{e} x^{\\frac{1}{n}} \\log x \\, dx -1 \\right)\r\n\\]\r\n\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u6c42\u3081\u308b\u7a4d\u5206\u3092 \\(I\\) \u3068\u304a\u304f.\r\n\\[\\begin{align}\r\nI & = \\left[ \\dfrac{n x^{\\frac{n+1}{n}}}{n+1} \\log x \\right] _ 1^e -\\dfrac{n}{n+1} \\displaystyle\\int _ {1}^{e} x^{\\frac{1}{n}} \\, dx \\\\\r\n& = \\dfrac{n e^{\\frac{n+1}{n}}}{n+1} -\\dfrac{n}{n+1} \\left[ \\dfrac{n x^{\\frac{n+1}{n}}}{n+1} \\right] _ 1^e \\\\\r\n& = \\dfrac{n}{(n+1)^2} \\left\\{ (n+1) e^{\\frac{n+1}{n}} -n \\left( e^{{\\frac{n+1}{n}}} -1 \\right) \\right\\} \\\\\r\n& = \\underline{\\dfrac{n \\left( e^{\\frac{n+1}{n}} +n \\right)}{(n+1)^2}}\r\n\\end{align}\\]\r\n (2)<\/strong>\r\n\\[\\begin{align}\r\nn (I-1) & = n \\left\\{ \\dfrac{n \\left( e^{\\frac{n+1}{n}} +n \\right)}{(n+1)^2} -1 \\right\\} \\\\\r\n& = \\dfrac{n^2 \\left( e^{\\frac{n+1}{n}} +n \\right) -n (n+1)^2}{(n+1)^2} \\\\\r\n& = \\dfrac{n^2 e^{1+\\frac{1}{n}} -2n^2 -n}{(n+1)^2} \\\\\r\n& = \\dfrac{e^{1+\\frac{1}{n}} -2 -\\frac{1}{n}}{\\left( 1+\\frac{1}{n} \\right)^2} \\\\\r\n& \\rightarrow e-2 \\quad ( \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d} )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} n \\left( \\displaystyle\\int _ {1}^{e} x^{\\frac{1}{n}} \\log x \\, dx -1 \\right) = \\underline{e-2}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u5b9a\u7a4d\u5206 \\[ \\displaystyle\\int _ {1}^{e} x^{\\frac{1}{n}} \\log x \\, dx \\quad ( n =1, 2, 3, \\cdots ) \\] … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n