{"id":6737,"date":"2021-07-21T00:10:28","date_gmt":"2021-07-20T15:10:28","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6737"},"modified":"2022-03-28T17:33:57","modified_gmt":"2022-03-28T08:33:57","slug":"integral-formulas","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/integral-formulas\/","title":{"rendered":"\uc801\ubd84 \uacf5\uc2dd"},"content":{"rendered":"
\n

\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 6\uc7a5 3\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.  (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n

\ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac\ub294 \ubbf8\ubd84\uc758 \uc5ed\uc5f0\uc0b0\uc73c\ub85c \uc815\uc801\ubd84\uc744 \uacc4\uc0b0\ud560 \uc218 \uc788\uac8c \ud574\uc900\ub2e4. \ub610\ud55c \ubd80\uc815\uc801\ubd84\uc740 \uadf8 \uc790\uccb4\ub85c \ubbf8\ubd84\uc758 \uc5ed\uc5f0\uc0b0\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc5ec\ub7ec \uac00\uc9c0 \uc801\ubd84 \uacf5\uc2dd\uc740 \ubbf8\ubd84 \uacf5\uc2dd\uc73c\ub85c\ubd80\ud130 \uc5bb\uc5b4\uc9c4\ub2e4.<\/p>\n

<\/p>\n

\uae30\ubcf8 \uacf5\uc2dd<\/h2>\n

\uae30\ubcf8 \ud568\uc218\uc758 \ubbf8\ubd84\uacf5\uc2dd\uc73c\ub85c\ubd80\ud130 \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

\n

\ubcf4\uae30 6.3.1.<\/span>
\n\ub2e4\uc74c \uc608\uc5d0\uc11c \\(C\\)\ub294 \uad6d\uc18c\uc801 \uc0c1\uc218\uc774\ub2e4.<\/p>\n

\\(\\displaystyle (1) \\,\\, \\int k \\,dx = kx+C \\quad \\text{if \\(k\\) is a constant.}\\)<\/p>\n

\\(\\displaystyle (2) \\,\\, \\int x^{\\alpha} dx = \\frac{1}{\\alpha +1} x^{\\alpha +1} \\quad \\text{if } \\alpha \\ne 0 ,\\,\\, \\alpha \\ne -1. \\)<\/p>\n

\\(\\displaystyle (3) \\,\\, \\int \\sin x \\,dx = – \\cos x +C . \\)<\/p>\n

\\(\\displaystyle (4) \\,\\, \\int \\cos x \\,dx = \\sin x +C. \\)<\/p>\n

\\(\\displaystyle (5) \\,\\, \\int \\sec^2 x \\,dx = \\tan x +C. \\)<\/p>\n

\\(\\displaystyle (6) \\,\\, \\int e^x \\,dx = e^x +C. \\)<\/p>\n

\\(\\displaystyle (7) \\,\\, \\int a^x \\,dx = \\frac{a^x}{\\ln a} +C \\quad \\text{if } a > 0 ,\\,\\, a\\ne 1. \\)<\/p>\n

\\(\\displaystyle (8) \\,\\, \\int \\frac{1}{x} dx = \\ln x +C. \\)<\/p>\n<\/div>\n

\n

\ucc38\uace0.<\/span>
\n\uc704 \ubcf4\uae30\uc5d0\uc11c \\(C\\)\ub97c \ub2e8\uc21c\ud788 \uc0c1\uc218\ub77c\uace0 \ud558\uc9c0 \uc54a\uace0 \uad6d\uc18c\uc801 \uc0c1\uc218\ub77c\uace0 \ud558\ub294 \ub370\uc5d0\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \uc774\uc720\uac00 \uc788\ub2e4. (8)\uc744 \uc774\uc6a9\ud558\uc5ec \uc124\uba85\ud574 \ubcf4\uaca0\ub2e4. \\(x\\)\uac00 \uc18d\ud55c \ubc94\uc704\uac00 \uc591\uc218\uc640 \uc74c\uc218\ub97c \ubaa8\ub450 \uc6d0\uc18c\ub85c \uac00\uc9c4\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0
\n\\[f(x) = \\begin{cases}
\n\\ln \\lvert x \\rvert +1 \\quad &\\text{if} \\,\\, x < 0 , \\\\[5pt]\n\\ln \\lvert x \\rvert +2 \\quad &\\text{if} \\,\\, x > 0
\n\\end{cases}\\]
\n\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(0\\)\uc774 \uc544\ub2cc \uc784\uc758\uc758 \\(x\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[f ‘ (x) = \\frac{1}{x}\\]
\n\uc774\ubbc0\ub85c \\(f(x)\\)\ub294 \\(1\/x\\)\uc758 \ubd80\uc815\uc801\ubd84\uc774\ub2e4. \uc774\uc640 \uac19\uc774 \\(1\/x\\)\uc758 \ubd80\uc815\uc801\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uaf34\ub85c \ud45c\ud604\ub41c\ub2e4.
\n\\[\\int \\frac{1}{x} dx = \\begin{cases}
\n\\ln \\lvert x \\rvert + C_1 \\quad &\\text{if} \\,\\, x < 0 , \\\\[5pt]\n\\ln \\lvert x \\rvert + C_2 \\quad &\\text{if} \\,\\, x > 0
\n\\end{cases}\\]
\n\uc5ec\uae30\uc11c \\(C_1\\)\uacfc \\(C_2\\)\ub294 \uac01\uac01 \uc0c1\uc218\uc774\ub2e4. \uc774 \ub450 \uc0c1\uc218\ub294 \uc11c\ub85c \ub2e4\ub97c \uc218\ub3c4 \uc788\ub2e4. \ub9cc\uc57d \\(C\\)\uac00 \uad6d\uc18c\uc801 \uc0c1\uc218\ub77c\uba74
\n\\[\\int \\frac{1}{x}dx = \\ln \\lvert x \\rvert +C\\]
\n\uc640 \uac19\uc774 \uac04\ub2e8\ud558\uac8c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.
\n<\/span><\/p>\n<\/div>\n

<\/p>\n

\ubd80\ubd84\uc801\ubd84\ubc95<\/h2>\n

\ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ubbf8\ubd84\uc758 \uacf1 \ubc95\uce59\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\frac{d}{dx} (f(x)g(x)) = f ‘ (x) g(x) + f(x) g ‘ (x) .\\]
\n\uc591\ubcc0\uc758 \uc5ed\ub3c4\ud568\uc218\ub97c \uc0dd\uac01\ud558\uba74 \uc704 \uc2dd\uc73c\ub85c\ubd80\ud130 \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
\n\\[f(x)g(x) = \\int f ‘ (x) g(x) dx + \\int f(x) g ‘ (x) dx .\\]
\n\ub530\ub77c\uc11c \ub2e4\uc74c\uacfc \uac19\uc740 \ubd80\ubd84\uc801\ubd84\ubc95\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

\n

\uc815\ub9ac 6.3.1. (\ubd80\ubd84\uc801\ubd84\ubc95)<\/span><\/p>\n

\ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0, \\(f ‘ \\)\uacfc \\(g ‘ \\)\uc774 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uadf8\ub7ec\uba74
\n\\[\\int_a^b f(x)g ‘ (x) dx = f(b)g(b) – f(a)g(a) – \\int_a^b f ‘ (x)g(x)dx \\]
\n\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\ubcf4\uae30 6.3.2.<\/span><\/p>\n

\ub2e4\uc74c \uc815\uc801\ubd84\uc744 \uad6c\ud574 \ubcf4\uc790.
\n\\[\\int_1^e \\ln x\\,dx.\\]
\n\\(f(x) = \\ln x,\\) \\(g(x)=x\\)\ub77c\uace0 \ud558\uba74 \\(f ‘ (x) = \\frac{1}{x} ,\\) \\(g ‘ (x) = 1\\)\uc774\ub2e4.
\n\\[f(x) g ‘(x) = \\frac{d}{dx} (f(x)g(x)) – f ‘ (x) g (x)\\]
\n\uc774\ubbc0\ub85c
\n\\[\\ln x = \\frac{d}{dx} (\\ln x \\times x) – \\frac{1}{x} \\times x\\]
\n\uc774\uba70, \uc591\ubcc0\uc758 \uc5ed\ub3c4\ud568\uc218\ub97c \uad6c\ud558\uba74
\n\\[\\int \\ln x\\,dx = x\\ln x -x +C\\]
\n\ub97c \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
\n\\[\\int_1^e \\ln x\\,dx = \\bigg[ x\\ln x – x \\bigg]_1^e = (e-e) – (0-1) = 1\\]
\n\uc774\ub2e4.\n<\/p>\n<\/div>\n

\ubd80\ubd84\uc801\ubd84\ubc95\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ubd80\uc815\uc801\ubd84 \ud615\ud0dc\ub85c \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.
\n\\[\\int f(x) g’ (x) dx = f(x)g(x) – \\int f ‘ (x)g(x) dx.\\]
\n\uc5ec\uae30\uc11c \\(u=f(x),\\) \\(v=g(x)\\)\ub77c\uace0 \ud558\uba74
\n\\[\\frac{du}{dx} = f ‘ (x) ,\\quad \\frac{dv}{dx} = g'(x)\\]
\n\uc989
\n\\[du = f ‘ (x) dx ,\\quad dv = g ‘ (x) dx\\]
\n\uc774\ubbc0\ub85c \ubd80\uc815\uc801\ubd84\uc758 \ubd80\ubd84\uc801\ubd84\ubc95\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.
\n\\[\\int u \\,dv = uv – \\int v\\,du .\\]\n<\/p>\n

\n

\ubcf4\uae30 6.3.3.<\/span><\/p>\n

\ub2e4\uc74c \ubd80\uc815\uc801\ubd84\uc744 \uad6c\ud574 \ubcf4\uc790.
\n\\[\\int xe^x \\,dx.\\]
\n\\(u=x,\\) \\(dv = e^x dx\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(dv = v ‘ (x) dx \\) \uadf8\ub9ac\uace0 \\(dv = e^x dx\\)\uc774\ubbc0\ub85c \\(v ‘ (x) = e^x\\)\uc774\ub2e4. \ub530\ub77c\uc11c
\n\\[v = \\int e^x dx = e^x +C_1\\]
\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(v=e^x\\)\ub85c\uc11c \uac00\uc7a5 \uac04\ub2e8\ud55c \ud615\ud0dc\uc758 \uc5ed\ub3c4\ud568\uc218\ub97c \ucde8\ud558\uc790.<\/p>\n

\\(du = dx\\)\uc774\ubbc0\ub85c
\n\\[\\int u \\,dv = uv – \\int v\\,du\\]
\n\ub85c\ubd80\ud130
\n\\[\\int xe^x dx = xe^x – \\int e^x dx = xe^x -e^x +C\\]
\n\ub97c \uc5bb\ub294\ub2e4.\n<\/p>\n<\/div>\n

<\/p>\n

\uce58\ud658\uc801\ubd84\ubc95<\/h2>\n

\\(F\\)\uc640 \\(g\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub610\ud55c \ub2eb\ud78c\uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \\(F ‘ = f\\)\uac00 \uc131\ub9bd\ud55c\ub2e4\uace0 \ud558\uc790. \ubbf8\ubd84\uc758 \uc5f0\uc1c4 \ubc95\uce59\uc73c\ub85c\ubd80\ud130 \ub2e4\uc74c \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\frac{d}{dt} F(g(t)) = F ‘ (g(t)) g ‘ (t) = f(g(t)) g ‘ (t).\\]
\n\ub530\ub77c\uc11c
\n\\[\\int f(g(t)) f ‘ (t) dt = F(x) +C\\]
\n\uc774\ub2e4. \uadf8\ub7f0\ub370
\n\\[F(x) +C = \\int f(x)dx\\]
\n\uc774\ubbc0\ub85c
\n\\[\\int f(x) dx = \\int f(g(t)) g ‘ (t) dt\\]
\n\uc774\ub2e4.<\/p>\n

\n

\uc815\ub9ac 6.3.2. (\ubd80\uc815\uc801\ubd84\uc758 \ubcc0\uc218\ubcc0\ud658)<\/span><\/p>\n

\\(f\\)\uac00 \uc5f0\uc18d\ud568\uc218\uc774\uace0 \\(g\\)\uac00 \ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc774\uba74
\n\\[\\int f(x)dx = \\int f(g(t)) g ‘ (t) dt\\]
\n\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc608\uc81c 6.3.4.<\/span> \ub2e4\uc74c \ubd80\uc815\uc801\ubd84\uc744 \uad6c\ud558\uc2dc\uc624.
\n\\[\\int t(t^2 + 3 )^4 \\,dt .\\]<\/p>\n

\ud480\uc774.<\/span>
\n\\(f(x)=x^4 ,\\) \\(g(t) = t^2 +3\\)\uc774\ub77c\uace0 \ud558\uba74
\n\\[f(g(t)) = (t^2 + 3)^4 ,\\quad g ‘ (t)=2t\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ubcc0\uc218\ubcc0\ud658 \uacf5\uc2dd\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\int t(t^2 +3)^4 \\,dt
\n&= \\frac{1}{2} \\int f(g(t)) g ‘ (t) dt \\\\[5pt]
\n&= \\frac{1}{2} \\int f(x) dx \\\\[5pt]
\n&= \\frac{1}{2} \\int x^4 \\,dx \\\\[5pt]
\n&= \\frac{1}{10} x^5 +C \\\\[5pt]
\n&= \\frac{1}{10} (t^2 +3)^5 +C .
\n\\end{align}\\]\n<\/p>\n<\/div>\n

\n

\uc608\uc81c 6.3.5.<\/span> \ub2e4\uc74c \ubd80\uc815\uc801\ubd84\uc744 \uad6c\ud558\uc2dc\uc624.
\n\\[\\int \\tan t \\,dt .\\]<\/p>\n

\ud480\uc774.<\/span>
\n\\(x=\\cos t\\)\ub77c\uace0 \ud558\uba74 \\(dx = -\\sin t \\,dt\\)\uc774\ub2e4.
\n\\[\\int \\frac{1}{x} dx = \\ln \\lvert x \\rvert + C(x)\\]
\n\uc774\uace0 \\(C(x)\\)\ub294 \\((-\\infty ,\\, 0)\\)\uacfc \\((0,\\, \\infty )\\)\uc5d0\uc11c \uac01\uac01 \uc0c1\uc218\uc774\ub2e4. \ub530\ub77c\uc11c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\int \\tan t \\,dt
\n&= – \\int \\frac{1}{\\cos t} (-\\sin t) dt \\\\[5pt]
\n&= – \\int \\frac{1}{x} dx \\\\[5pt]
\n&= – \\ln \\lvert x \\rvert +C(x) \\\\[5pt]
\n&= – \\ln \\lvert \\cos t \\rvert + C(\\cos t).
\n\\end{align}\\]
\n\ub9cc\uc57d \\(I = \\left( -\\frac{\\pi}{2} ,\\, \\frac{\\pi}{2} \\right)\\)\ub77c\uace0 \ud558\uba74 \\(t\\in I\\)\uc77c \ub54c \\(\\cos t > 0\\)\uc774\ubbc0\ub85c \\(I\\)\uc5d0\uc11c \\(\\tan t\\)\uc758 \ubd80\uc815\uc801\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\int \\tan t \\,dt = – \\ln \\lvert \\cos t \\rvert +C .\\]
\n\uc5ec\uae30\uc11c \\(C\\)\ub294 \uc0c1\uc218\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc608\uc81c 6.3.6.<\/span> \ub2e4\uc74c \ubd80\uc815\uc801\ubd84\uc744 \uad6c\ud558\uc2dc\uc624.
\n\\[\\int \\sin^3 x \\,\\cos ^2 x \\,dx.\\]<\/p>\n

\ud480\uc774.<\/span>
\n\\(u=\\cos x\\)\ub77c\uace0 \ud558\uba74 \\(du = -\\sin x\\)\uc774\ub2e4. \ub530\ub77c\uc11c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\int \\sin^3 x \\,\\cos^2 x \\,dx
\n&= \\int \\sin^2 x\\,\\cos^2 x\\,\\sin x\\,dx \\\\[5pt]
\n&= \\int (1-\\cos^2 x ) \\cos^2 x \\,\\sin x\\,dx \\\\[5pt]
\n&= – \\int (1-u^2 )u^2 \\,du \\\\[5pt]
\n&= \\int (u^4 – u^2 ) du \\\\[5pt]
\n&= \\frac{1}{5} u^5 – \\frac{1}{3} u^3 +C \\\\[5pt]
\n&= \\frac{1}{5} \\cos^5 x – \\frac{1}{3} \\cos^3 x +C.
\n\\end{align}\\]\n<\/p>\n<\/div>\n

\n

\uc608\uc81c 6.3.7.<\/span> \ub2e4\uc74c \ubd80\uc815\uc801\ubd84\uc744 \uad6c\ud558\uc2dc\uc624.
\n\\[\\int \\sqrt{9-x^2} \\,dx \\quad (-3 \\le x \\le 3).\\]<\/p>\n

\ud480\uc774.<\/span>
\n\\(x=3\\cos t\\)\ub77c\uace0 \ud558\uba74 \\(dx = -3 \\sin t\\,dt\\)\uc774\uace0
\n\\[9-x^2 = 9 – 9 \\cos^2 t = (3\\sin t)^2\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\int\\sqrt{9-x^2} dx
\n&= \\int\\sqrt{(3\\sin t)^2} (-3 \\sin t)dt\\\\[5pt]
\n&= – 9 \\int \\sin^2 t\\,dt \\\\[5pt]
\n&= – \\frac{9}{2} \\int (1-\\cos 2t) dt\\\\[5pt]
\n&= – \\frac{9}{2} \\left( t – \\frac{1}{2} \\sin 2t \\right) +C.
\n\\end{align}\\]
\n\uc5ec\uae30\uc11c \\(t=\\cos^{-1} \\frac{x}{3}\\)\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\uc608\uc81c 6.3.8.<\/span> \ub2e4\uc74c \ubd80\uc815\uc801\ubd84\uc744 \uad6c\ud558\uc2dc\uc624.
\n\\[\\int \\frac{1}{x^2 -4} dx \\quad (-2 < x < 2) .\\]<\/p>\n

\ud480\uc774.<\/span>
\n\ud53c\uc801\ubd84\ud568\uc218\ub97c \ubcc0\ud615\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\frac{1}{x^2 -4} = \\frac{1}{(x+2)(x-2)} = \\frac{1}{4(x-2)} – \\frac{1}{4(x+2)}.\\]
\n\uadf8\ub7ec\ubbc0\ub85c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.
\n\\[\\begin{align}
\n\\int \\frac{1}{x^2 -4} dx
\n&= \\frac{1}{4} \\int \\left( \\frac{1}{x-2} – \\frac{1}{x+2} \\right)dx \\\\[5pt]
\n&= \\frac{1}{4} \\left( \\ln\\lvert x-2 \\rvert – \\ln \\lvert x+2 \\rvert \\right)+C.
\n\\end{align}\\]\n<\/p>\n<\/div>\n


\n<\/p>\n

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