\u6b21\u306e\u5b9a\u7a4d\u5206\u3092\u6c42\u3081\u3088.<\/p>\r\n
(1)<\/strong>\u3000\\(\\displaystyle\\int _ 0^{\\frac{\\pi}{4}} \\dfrac{dx}{1 +\\sin x}\\)<\/p><\/li>\r\n (2)<\/strong>\u3000\\(\\displaystyle\\int _ {\\frac{4}{3}}^{2} \\dfrac{dx}{x^2 \\sqrt{x-1}}\\)<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\\begin{align}\r\n\\dfrac{1}{1 +\\sin x} & = \\dfrac{1 -\\sin x}{\\cos^2 x} \\\\\r\n& = \\left( \\tan x +\\dfrac{1}{\\cos x} \\right)'\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\n\\displaystyle\\int _ 0^{\\frac{\\pi}{4}} \\dfrac{dx}{1 +\\sin x} & = \\left[ \\tan x +\\dfrac{1}{\\cos x} \\right] _ 0^{\\frac{\\pi}{4}} \\\\\r\n& = 1 +\\sqrt{2} -1 \\\\\r\n& = \\underline{\\sqrt{2}}\r\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \u6c42\u3081\u308b\u7a4d\u5206\u5024\u3092 \\(I\\) \u3068\u304a\u304f.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n
\r\n\\(t = \\sqrt{x-1}\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nx & = t^2 +1 \\\\\r\n\\text{\u2234} \\quad dx & = 2t \\, dt\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\r\n\\begin{array}{c|ccc} x & \\dfrac{4}{3} & \\rightarrow & 2 \\\\ \\hline t & \\dfrac{1}{\\sqrt{3}} & \\rightarrow & 1 \\end{array}\r\n\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nI & = \\displaystyle\\int _ \\frac{1}{\\sqrt{3}}^1 \\dfrac{2t}{(t^2+1)^2 t} \\, dt \\\\\r\n& = 2 \\displaystyle\\int _ \\frac{1}{\\sqrt{3}}^1 \\dfrac{dt}{(t^2+1)^2}\r\n\\end{align}\\]\r\n\u3055\u3089\u306b, \\(t = \\tan \\theta \\ \\left( -\\dfrac{\\pi}{2} \\lt \\theta \\lt \\dfrac{\\pi}{2} \\right)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\ndt = \\dfrac{d \\theta}{\\cos^2 \\theta}\r\n\\]\r\n\u307e\u305f\r\n\\[\r\n\\begin{array}{c|ccc} t & \\dfrac{1}{\\sqrt{3}} & \\rightarrow & 1 \\\\ \\hline \\theta & \\dfrac{\\pi}{6} & \\rightarrow & \\dfrac{\\pi}{4} \\end{array}\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\nI & = 2 \\displaystyle\\int _ {\\frac{\\pi}{6}}^{\\frac{\\pi}{4}} \\cos^2 \\theta \\, d \\theta \\\\\r\n& = \\displaystyle\\int _ {\\frac{\\pi}{6}}^{\\frac{\\pi}{4}} \\left( 1 +\\cos 2 \\theta \\right) \\, d \\theta \\\\\r\n& = \\left[ \\theta +\\dfrac{\\sin 2 \\theta}{2} \\right] _ {\\frac{\\pi}{6}}^{\\frac{\\pi}{4}} \\\\\r\n& = \\left( \\dfrac{\\pi}{4} +\\dfrac{1}{2} \\right) -\\left( \\dfrac{\\pi}{6} +\\dfrac{\\sqrt{3}}{4} \\right) \\\\\r\n& = \\underline{\\dfrac{\\pi}{12} +\\dfrac{2 -\\sqrt{3}}{4}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u5b9a\u7a4d\u5206\u3092\u6c42\u3081\u3088. (1)\u3000\\(\\displaystyle\\int _ 0^{\\frac{\\pi}{4}} \\dfrac{dx}{1 +\\sin x}\\) (2)\u3000\\(\\displaystyle\\int _ {\\fra … \u7d9a\u304d\u3092\u8aad\u3080