\u591a\u9805\u5f0f\u306e\u5e73\u65b9\u6839\u306a\u3069\u306e\u7121\u7406\u95a2\u6570\u3092\u542b\u3080\u5834\u5408\u306e\u7a4d\u5206\u3002\u5fc5\u305a\u7a4d\u5206\u3067\u304d\u308b\u3068\u3044\u3046\u308f\u3051\u3067\u306f\u306a\u3044\u304c\uff0c\u7f6e\u63db\u7a4d\u5206\u306a\u3069\u306b\u3088\u3063\u3066\u7a4d\u5206\u3067\u304d\u308b\u3044\u304f\u3064\u304b\u306e\u4f8b\u3092\u3042\u3052\u3066\u304a\u304f\u3002\uff08\u7121\u7406\u95a2\u6570\u306e\u7a4d\u5206\u306f\u7121\u7406\u3067\u3059\uff01\u306a\u3093\u3066\u8a00\u308f\u305a\u306b… \uff09<\/p>\n
\\( \\sqrt{x+1} = t\\) \u3068\u304a\u304f\u3068\uff0c<\/p>\n
\\begin{eqnarray}
\nx + 1 &=& t^2 \\\\
\nx &=& t^2 -1 \\\\
\ndx &=& 2t dt
\n\\end{eqnarray}<\/p>\n
\u3057\u305f\u304c\u3063\u3066\uff0c<\/p>\n
\\begin{eqnarray}
\n\\int \\frac{dx}{x \\sqrt{x+1}} &=& \\int \\frac{2 t dt}{(t^2 -1) t} \\\\
\n&=& \\int \\frac{2}{t^2-1} dt\\\\
\n&=& \\int \\left( \\frac{1}{t-1} -\\frac{1}{t+1}\\right) dt\\\\
\n&=& \\log |t-1| -\\log |t + 1| + C \\\\
\n&=& \\log \\left| \\frac{\\sqrt{x+1} -1}{\\sqrt{x+1} + 1}\\right| + C
\n\\end{eqnarray}<\/p>\n
\\( \\sqrt{x^2 + 1} = t\\) \u3067\u306f\u306a\u304f\uff0c\\( \\sqrt{x^2 + 1} = t {\\color{red} { -x}}\\) \u3068\u304a\u304f\u3068\u3088\u3044\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\sqrt{x^2+1} &=& t -x \\\\
\nx^2 + 1 &=& t^2 -2 t x + x^2 \\\\
\n\\therefore\\ \\ x &=& \\frac{t^2-1}{2t} \\\\
\ndx &=& \\frac{t^2 + 1}{2t^2} dt \\\\
\n\\sqrt{x^2+1} &=& t -x \\\\
\n&=& t -\\frac{t^2 -1}{2t} \\\\
\n&=& \\frac{t^2 + 1}{2t}
\n\\end{eqnarray}<\/p>\n
\u3057\u305f\u304c\u3063\u3066<\/p>\n
\\begin{eqnarray}
\n\\int \\frac{dx}{\\sqrt{x^2+1}} &=& \\int \\frac{2t}{t^2 + 1} \\frac{t^2 + 1}{2t^2} dt\\\\
\n&=& \\int \\frac{1}{t} dt \\\\
\n&=& \\log |t| + C \\\\
\n&=& \\log \\left|x + \\sqrt{x^2 + 1}\\right| + C \\\\
\n&=& \\log \\left(x + \\sqrt{x^2 + 1}\\right) + C
\n\\end{eqnarray}<\/p>\n
\u9006\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u5fae\u5206\u3092\u601d\u3044\u51fa\u3059\u3068<\/p>\n
$$ \\frac{d}{dx} \\sinh^{-1} x = \\frac{d}{dx} \\log\\left(x + \\sqrt{x^2+1}\\right) = \\frac{1}{\\sqrt{x^2+1}}$$<\/p>\n
\u3060\u3063\u305f\u304b\u3089\uff0c\u305f\u3060\u3061\u306b<\/p>\n
$$\\int\\frac{dx}{\\sqrt{x^2+1}} = \\sinh^{-1} x + C = \\log\\left(x + \\sqrt{x^2+1}\\right) + C$$<\/p>\n
\u3053\u306e\u5834\u5408\u3082 \\( \\sqrt{x^2 + 1} = t -x\\) \u3068\u304a\u304f\u3068\u3088\u3044\u3053\u3068\u304c\u77e5\u3089\u308c\u3066\u3044\u308b\u3002<\/p>\n
\\begin{eqnarray}
\n\\int \\sqrt{x^2+1} dx &=& \\int \\frac{t^2+1}{2t} \\frac{t^2 + 1}{2t^2} dt\\\\
\n&=& \\frac{1}{4} \\int \\frac{t^4 + 2 t^2 + 1}{t^3} dt \\\\
\n&=& \\frac{1}{4} \\int \\left(t + \\frac{2}{t} + \\frac{1}{t^3} \\right)dt\\\\
\n&=& \\frac{1}{4} \\left( \\frac{t^2}{2} + 2 \\log |t| -\\frac{1}{2 t^2} \\right) + C \\\\
\n&=& \\frac{1}{2} \\log |t| + \\frac{1}{2} \\frac{t^2 + 1}{2t} \\frac{t^2 -1}{2t} + C\\\\
\n&=& \\frac{1}{2} \\log \\left(x + \\sqrt{x^2+1}\\right)+ \\frac{1}{2} x \\sqrt{x^2+1}+ C \\\\
\n&=& \\frac{1}{2} \\sinh^{-1} x + \\frac{1}{2} x \\sqrt{x^2+1}+ C
\n\\end{eqnarray}<\/p>\n
<\/p>\n
\u90e8\u5206\u7a4d\u5206\u3092\u3057\u3066\uff0c\u4f8b 2 \u306e\u7d50\u679c\u3092\u4f7f\u3046\u3002<\/p>\n
\\begin{eqnarray}
\n\\int \\sqrt{x^2+1} dx &=& x \\sqrt{x^2 + 1} -\\int x \\frac{x}{\\sqrt{x^2+1}} dx \\\\
\n&=& x \\sqrt{x^2 + 1} -\\int\u00a0 \\frac{x^2 + 1}{\\sqrt{x^2+1}} dx + \\int\u00a0 \\frac{1}{\\sqrt{x^2+1}} dx\\\\
\n&=& \\int\u00a0 \\frac{1}{\\sqrt{x^2+1}} dx + x \\sqrt{x^2 + 1} -\\int \\sqrt{x^2 + 1} dx \\\\ \\ \\\\
\n\\therefore\\ \\ \\int \\sqrt{x^2+1} dx &=& \\frac{1}{2} \\left\\{ \\int\u00a0 \\frac{1}{\\sqrt{x^2+1}} dx + x \\sqrt{x^2 + 1}\\right\\} \\\\
\n&=& \\frac{1}{2} \\log \\left(x + \\sqrt{x^2+1}\\right) + \\frac{1}{2} x \\sqrt{x^2+1}+ C \\\\
\n&=& \\frac{1}{2} \\sinh^{-1} x + \\frac{1}{2} x \\sqrt{x^2+1}+ C
\n\\end{eqnarray}<\/p>\n
$x = \\sin t, \\ dx = \\cos t\\, dt$ \u3068\u3057\u3066<\/p>\n
\\begin{eqnarray}
\n\\int \\sqrt{1 -x^2}\\, dx &=& \\int \\sqrt{1 -\\sin^2 t} \\, \\cos t\\, dt \\\\
\n&=& \\int \\cos^2 t\\, dt \\\\
\n&=& \\frac{1}{2} \\int (\\cos 2 t +1)\\, dt \\\\
\n&=& \\frac{1}{2} \\left(\\frac{1}{2} \\sin 2 t + t\\right) \\\\
\n&=& \\frac{1}{2} \\left(\\sin\u00a0 t \\cos t + t\\right) \\\\
\n&=& \\frac{1}{2} \\left(x \\sqrt{1 -x^2} + \\sin^{-1} x \\right)
\n\\end{eqnarray}<\/p>\n
\\begin{eqnarray}
\nI &\\equiv& \\int \\sqrt{1-x^2}\\, dx \\\\
\n&=& x \\sqrt{1 -x^2} -\\int x\\, \\left(\\frac{d}{dx} \\sqrt{1 -x^2}\\right) \\,dx\\\\
\n&=& x \\sqrt{1-x^2} + \\int \\frac{x^2}{\\sqrt{1-x^2}}\\,dx \\\\
\n&=& x \\sqrt{1-x^2} + \\int \\frac{1}{\\sqrt{1-x^2}} \\,dx -\\int \\frac{1-x^2}{\\sqrt{1-x^2}}\\, dx \\\\
\n&=& x \\sqrt{1-x^2} + \\int \\frac{1}{\\sqrt{1-x^2}} \\,dx -I \\\\
\n\\therefore\\ \\ I &=& \\frac{1}{2} \\left(x \\sqrt{1-x^2} + \\int \\frac{1}{\\sqrt{1-x^2}} \\,dx \\right) \\\\
\n&=& \\frac{1}{2}\\left(x\\sqrt{1-x^2} +\\sin^{-1} x\\right)
\n\\end{eqnarray}<\/p>\n
$\\sqrt{x-1} = t$\u00a0 \u3068\u304a\u304f\u3068…<\/p>\n
\u3053\u306e\u7a4d\u5206\u306f\u76f8\u5bfe\u8ad6\u7684\u5b87\u5b99\u8ad6\u306e\u307b\u3046\u3067\u5fc5\u8981\u306b\u306a\u3063\u305f\u308a\u3059\u308b<\/a>\u3093\u3067\u3059\u3088\u3002<\/p>\n \u3053\u3093\u306a\u7a4d\u5206\uff0c\u793e\u4f1a\u306b\u51fa\u305f\u3089\u7d76\u5bfe\u4f7f\u308f\u306a\u3044\u3057\uff0c\u5f79\u306b\u306a\u3093\u304b\u305f\u305f\u306a\u3044\u306a\u3069\u3068\u601d\u3063\u3066\u3044\u307e\u305b\u3093\u304b\uff1f\u3053\u308c\u3089\u306e\u7a4d\u5206\u3082\u5b87\u5b99\u5e74\u9f62\u3092\u77e5\u308b\u305f\u3081\u306b\u5fc5\u8981\u306b\u306a\u3063\u305f\u308a\u3059\u308b<\/a>\u3093\u3067\u3059\u3088\u3002<\/p>\n \u3053\u3093\u306a\u7a4d\u5206\uff0c\u3044\u3063\u305f\u3044\u3044\u3064\u3069\u3053\u3067\u4f7f\u3046\u3093\u3060\u3068\u601d\u3063\u3066\u3044\u307e\u305b\u3093\u304b\uff1f\u3053\u306e\u7a4d\u5206\u306f1\u5e74\u751f\u5f8c\u671f\u306e\u96fb\u78c1\u6c17\u5b66I\u3067\u4e00\u69d8\u306a\u7dda\u96fb\u8377\u306b\u3088\u308b\u96fb\u5834<\/a>\u3092\u6c42\u3081\u308b\u969b\u306b\u4f7f\u3046\u3093\u3067\u3059\u3088\uff01<\/p>\n","protected":false},"excerpt":{"rendered":" \u591a\u9805\u5f0f\u306e\u5e73\u65b9\u6839\u306a\u3069\u306e\u7121\u7406\u95a2\u6570\u3092\u542b\u3080\u5834\u5408\u306e\u7a4d\u5206\u3002\u5fc5\u305a\u7a4d\u5206\u3067\u304d\u308b\u3068\u3044\u3046\u308f\u3051\u3067\u306f\u306a\u3044\u304c\uff0c\u7f6e\u63db\u7a4d\u5206\u306a\u3069\u306b\u3088\u3063\u3066\u7a4d\u5206\u3067\u304d\u308b\u3044\u304f\u3064\u304b\u306e\u4f8b\u3092\u3042\u3052\u3066\u304a\u304f\u3002\uff08\u7121\u7406\u95a2\u6570\u306e\u7a4d\u5206\u306f\u7121\u7406\u3067\u3059\uff01\u306a\u3093\u3066\u8a00\u308f\u305a\u306b… \uff09<\/p>\u7df4\u7fd2\u554f\u984c 2. \\(\\displaystyle \\int \\frac{\\sqrt{x}}{\\sqrt{1-x}} dx \\)<\/h3>\n
\u7df4\u7fd2\u554f\u984c 3. \\(\\displaystyle \\int \\frac{\\sqrt{x}}{\\sqrt{1-x^3}} dx \\)<\/h3>\n
\u7df4\u7fd2\u554f\u984c 4. \\(\\displaystyle \\int \\frac{1}{\\left(a^2 +x^2\\right)^{\\frac{3}{2}}} dx \\)<\/h3>\n