{"id":6724,"date":"2021-07-21T00:06:25","date_gmt":"2021-07-20T15:06:25","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6724"},"modified":"2022-03-06T19:43:23","modified_gmt":"2022-03-06T10:43:23","slug":"antiderivatives","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/antiderivatives\/","title":{"rendered":"\uc5ed\ub3c4\ud568\uc218\uc640 \ubd80\uc815\uc801\ubd84"},"content":{"rendered":"
\n

\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 5\uc7a5 5\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.  (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n

<\/p>\n

\uc5ed\ub3c4\ud568\uc218\uc640 \ubd80\uc815\uc801\ubd84<\/h2>\n

\\(I\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774\uace0 \\(F\\)\uc640 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud558\uc790. \ub9cc\uc57d \\(F\\)\uac00 \\(I\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0, \\(I\\)\uc758 \uc784\uc758\uc758 \uc810 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(F ‘ (x) = f(x)\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\uba74 \\(F\\)\ub97c \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \uc5ed\ub3c4\ud568\uc218<\/span>(antiderivative)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\ud55c \ud568\uc218\uc758 \uc5ed\ub3c4\ud568\uc218\uac00 \ud558\ub098\ub85c \uc815\ud574\uc9c0\uc9c0\ub294 \uc54a\ub294\ub2e4. \uc608\ucee8\ub300
\n\\[f(x)= \\cos x\\]
\n\uc774\uba74
\n\\[\\begin{align}
\nF_1 (x) &= \\sin x , \\\\[5pt]
\nF_2 (x) &= \\sin x + 4 ,\\\\[5pt]
\nF_3 (x) &= \\sin x – \\pi
\n\\end{align}\\]
\n\ub294 \ubaa8\ub450 \\(f\\)\uc758 \uc5ed\ub3c4\ud568\uc218\ub97c \uc815\uc758\ud55c\ub2e4.<\/p>\n

\uc77c\ubc18\uc801\uc73c\ub85c, \\(F_1\\)\uacfc \\(F_2\\)\uc774 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \uc5ed\ub3c4\ud568\uc218\uc774\uba74 \\(I\\)\uc758 \ubaa8\ub4e0 \uc810 \\(x\\)\uc5d0 \ub300\ud558\uc5ec \\(F_1 ‘ (x) = F_2 ‘ (x)\\)\uc774\ubbc0\ub85c \\(F_1\\)\uacfc \\(F_2\\)\ub294 \\(I\\)\uc5d0\uc11c \uc0c1\uc218 \ucc28\uc774\uc774\ub2e4.<\/p>\n

\n

\uc815\ub9ac 5.5.1. (\uc5ed\ub3c4\ud568\uc218\uc758 \uc77c\ubc18\ud615)<\/span><\/p>\n

\ud568\uc218 \\(F\\)\uac00 \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ud568\uc218 \\(f\\)\uc758 \ud55c \uc5ed\ub3c4\ud568\uc218\uc774\uba74, \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \uc5ed\ub3c4\ud568\uc218\uc758 \uc77c\ubc18\ud615(general antiderivative)\uc740
\n\\[F(x) +C\\]
\n\uaf34\uc774\ub2e4. \uc5ec\uae30\uc11c \\(C\\)\ub294 \uc784\uc758\uc758 \uc0c1\uc218\uc774\ub2e4.\n<\/p>\n<\/div>\n

\n

\ucc38\uace0.<\/span>
\n\uc704 \uc815\ub9ac\uc5d0\uc11c \\(I\\)\uac00 \uad6c\uac04\uc774 \uc544\ub2c8\ub77c\uba74 \\(C\\)\ub294 \uad6d\uc18c\uc801 \uc0c1\uc218(locally constant)\uc774\ub2e4. \uc608\ucee8\ub300
\n\\[I_1 = [0,\\,1] ,\\,\\, I_2 = [2,\\,3] ,\\,\\, I = I_1 \\cup I_2\\]
\n\uc774\uace0
\n\\[\\begin{align}
\nf(x) &= 3x^2 ,\\\\[5pt]
\nF_1 (x) &= x^3 , \\\\[5pt]
\nF_2 (x) &= \\begin{cases} x^3 \\quad &\\text{if}\\,\\, x\\in I_1 ,\\\\[5pt] x^3 +1 \\quad &\\text{if}\\,\\, x\\in I_2 \\end{cases}
\n\\end{align}\\]
\n\ub77c\uace0 \ud558\uba74 \\(F_1\\)\uacfc \\(F_2\\)\ub294 \uc0c1\uc218 \ucc28\uc774\uac00 \uc544\ub2c8\uc9c0\ub9cc \ub450 \ud568\uc218 \ubaa8\ub450 \\(f\\)\uc758 \uc5ed\ub3c4\ud568\uc218\uc774\ub2e4. \ub9cc\uc57d \ud568\uc218 \\(C\\)\ub97c
\n\\[C (x) = \\begin{cases} 0 \\quad &\\text{if}\\,\\, x\\in I_1 ,\\\\[5pt] 1 \\quad &\\text{if}\\,\\, x\\in I_2 \\end{cases}\\]
\n\uc774\ub77c\uace0 \uc815\uc758\ud558\uba74 \\(C\\)\ub294 \uad6d\uc18c\uc801 \uc0c1\uc218\uc774\uace0
\n\\[F_2 (x) = F_1 (x) +C(x) \\]
\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\n

\ubcf4\uae30 5.5.1.<\/span><\/p>\n

\\(\\mathbb{R}\\)\uc5d0\uc11c \\(f(x)= \\sin x\\)\uc758 \uc5ed\ub3c4\ud568\uc218\uc758 \uc77c\ubc18\ud615\uc744 \uad6c\ud574 \ubcf4\uc790. \uba3c\uc800
\n\\[F_1 (x) = -\\cos x\\]
\n\ub77c\uace0 \ud558\uba74
\n\\[\\frac{d}{dx} F_1 (x) = \\frac{d}{dx}(-\\cos x)=\\sin x\\]
\n\uc774\ubbc0\ub85c \\(F\\)\ub294 \\(f\\)\uc758 \ud55c \uc5ed\ub3c4\ud568\uc218\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f\\)\uc758 \uc5ed\ub3c4\ud568\uc218\uc758 \uc77c\ubc18\ud615\uc740
\n\\[F(x) = -\\cos x +C\\]
\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(C\\)\ub294 \uc784\uc758\uc758 \uc0c1\uc218\uc774\ub2e4.\n<\/p>\n<\/div>\n

\ud568\uc218 \\(f\\)\uc758 \ubaa8\ub4e0 \uc5ed\ub3c4\ud568\uc218\ub4e4\uc758 \ubaa8\uc784\uc744 \\(f\\)\uc758 \ubd80\uc815\uc801\ubd84<\/span>(indefinite integral)\uc774\ub77c\uace0 \ubd80\ub974\uace0
\n\\[\\int f(x) dx\\]
\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \uc5ec\uae30\uc11c \\(\\int\\)\ub294 \uc801\ubd84\uae30\ud638<\/span>(integral sign)\uc774\ub2e4. \uc704\uc640 \uac19\uc740 \ud45c\ud604\uc5d0\uc11c \\(f\\)\ub97c \ud53c\uc801\ubd84\ud568\uc218<\/span>(integrand)\ub77c\uace0 \ubd80\ub974\uba70, \\(x\\)\ub97c \uc801\ubd84\ubcc0\uc218<\/span>(variable of integration)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\n

\ubcf4\uae30 5.5.2.<\/span>
\n\ub2e4\uc74c \uc608\uc5d0\uc11c \\(C\\)\ub294 \uad6d\uc18c\uc801 \uc0c1\uc218\uc774\ub2e4.<\/p>\n

    \n
  1. \\(\\displaystyle \\int 3x \\,dx = \\frac{3}{2} x^2 +C , \\)<\/li>\n
  2. \\(\\displaystyle \\int \\sin x \\,dx = -\\cos x +C , \\)<\/li>\n
  3. \\(\\displaystyle \\int \\left( e^t + \\sec^2 t + \\frac{1}{t} \\right) \\,dt = e^t + \\tan t – \\frac{1}{t^2} +C . \\)<\/li>\n<\/ol>\n<\/div>\n

    <\/p>\n

    \ubbf8\ubd84\ubc29\uc815\uc2dd\uacfc \ucd08\uae43\uac12 \ubb38\uc81c<\/h2>\n

    \ud568\uc218 \\(f(x)\\)\uc758 \uc5ed\ub3c4\ud568\uc218\ub97c \uad6c\ud558\ub294 \uac83\uc740 \ub4f1\uc2dd
    \n\\[\\frac{dy}{dx} = f(x)\\]
    \n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \ud568\uc218 \\(y(x)\\)\ub97c \uad6c\ud558\ub294 \uac83\uacfc \uac19\ub2e4. \uc774\ub7ec\ud55c \ubc29\uc815\uc2dd\uc744 \ubbf8\ubd84\ubc29\uc815\uc2dd<\/span>(differential equation)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc704\uc640 \uac19\uc740 \ud615\ud0dc\uc758 \ubbf8\ubd84\ubc29\uc815\uc2dd\uc744 \ud480\uae30 \uc704\ud574\uc11c\ub294 \\(f(x)\\)\uc758 \uc5ed\ub3c4\ud568\uc218\ub97c \uad6c\ud574\uc57c \ud55c\ub2e4. \ubbf8\ubd84\ubc29\uc815\uc2dd\uc744 \ud478\ub294 \uacfc\uc815\uc5d0\uc11c \ub098\ud0c0\ub098\ub294 \uc0c1\uc218\ub97c
    \n\\[y \\left( x_0 \\right) = y_0\\]
    \n\uc640 \uac19\uc774 \uc815\ud560 \uc218 \uc788\ub294\ub370, \uc774\uc640 \uac19\uc774 \uc0c1\uc218\ub97c \uc815\ud558\ub294 \uc870\uac74\uc744 \ucd08\uae30\uc870\uac74<\/span>(initial condition)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ucd08\uae30\uc870\uac74\uc774 \uc8fc\uc5b4\uc9c4 \ubbf8\ubd84\ubc29\uc815\uc2dd\uc744 \ucd08\uae43\uac12 \ubb38\uc81c<\/span>(initial value problem)\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

    \n

    \ubcf4\uae30 5.5.3.<\/span>
    \n\ub2e4\uc74c \ucd08\uae43\uac12 \ubb38\uc81c\uc758 \ud574\ub97c \uad6c\ud558\uc2dc\uc624.<\/p>\n

      \n
    1. \\(\\displaystyle \\frac{dy}{dx} = 2x-7 ,\\quad y(2) =0.\\)<\/li>\n
    2. \\(\\displaystyle \\frac{ds}{dt} = 1+\\cos t ,\\quad s(\\pi) =1.\\)<\/li>\n
    3. \\(\\displaystyle \\frac{d^2 s}{dt^2} = \\frac{3t}{8} ,\\quad \\left. \\frac{ds}{dt} \\right\\vert_{t=4} = 3 ,\\quad s(4)=4.\\)<\/li>\n<\/ol>\n

      \ud480\uc774.<\/span><\/p>\n

        \n
      1. \ub9cc\uc57d \\(f(x) = x^2 -7x\\)\uc774\uba74 \\(f ‘ (x) = 2x-7\\)\uc774\ubbc0\ub85c \\(2x-7\\)\uc758 \uc5ed\ub3c4\ud568\uc218\uc758 \uc77c\ubc18\ud615\uc740
        \n\\[y = x^2 – 7x +C\\]
        \n\uaf34\uc774\ub2e4. \uc5ec\uae30\uc11c \\(C\\)\ub294 \uc0c1\uc218\uc774\ub2e4. \\(x=2,\\) \\(y=0\\)\uc744 \ub300\uc785\ud558\uba74
        \n\\[0=2^2 – 14 +C\\]
        \n\uc774\ubbc0\ub85c \\(C=10\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc8fc\uc5b4\uc9c4 \ucd08\uae43\uac12 \ubb38\uc81c\uc758 \ud574\ub294 \\[y=x^2 – 7x +10\\]\uc774\ub2e4.\n<\/li>\n
      2. \\(1+\\cos t\\)\uc758 \uc5ed\ub3c4\ud568\uc218\uc758 \uc77c\ubc18\ud615\uc740
        \n\\[s = t- \\sin t +C\\]
        \n\uaf34\uc774\ub2e4. \uc5ec\uae30\uc11c \\(C\\)\ub294 \uc0c1\uc218\uc774\ub2e4. \\(t = \\pi,\\) \\(s=1\\)\uc744 \ub300\uc785\ud558\uba74
        \n\\[1 = \\pi – \\sin \\pi +C\\]
        \n\uc774\ubbc0\ub85c \\(C = 1-\\pi\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc8fc\uc5b4\uc9c4 \ucd08\uae43\uac12 \ubb38\uc81c\uc758 \ud574\ub294
        \n\\[s = t-\\sin t + 1 – \\pi\\]
        \n\uc774\ub2e4.\n<\/li>\n
      3. \\(3t\/8\\)\uc758 \uc5ed\ub3c4\ud568\uc218\uc758 \uc77c\ubc18\ud615\uc740
        \n\\[\\frac{ds}{dt} = \\frac{3t^2}{16} +C_1\\]
        \n\uaf34\uc774\ub2e4. \uc5ec\uae30\uc11c \\(C_1\\)\uc740 \uc0c1\uc218\uc774\ub2e4. \\(t=4,\\) \\(ds\/dt = 3\\)\uc744 \ub300\uc785\ud558\uba74
        \n\\[3 = 3+C_1\\]
        \n\uc774\ubbc0\ub85c \\(C_1 =0\\)\uc774\ub2e4. \ub530\ub77c\uc11c
        \n\\[\\frac{ds}{dt} = \\frac{3t^2}{16}\\]
        \n\uc774\ub2e4. \uc774 \ud568\uc218\uc758 \uc5ed\ub3c4\ud568\uc218\uc758 \uc77c\ubc18\ud615\uc744 \uad6c\ud558\uba74
        \n\\[s = \\frac{t^3}{16} +C_2\\]
        \n\uc774\ub2e4. \uc5ec\uae30\uc11c \\(C_2\\)\ub294 \uc0c1\uc218\uc774\ub2e4. \uc774 \uc2dd\uc5d0 \\(t=4,\\) \\(s=4\\)\ub97c \ub300\uc785\ud558\uba74 \\(C_2 =0\\)\uc744 \uc5bb\ub294\ub2e4.
        \n\uadf8\ub7ec\ubbc0\ub85c \uc8fc\uc5b4\uc9c4 \ucd08\uae43\uac12 \ubb38\uc81c\uc758 \ud574\ub294
        \n\\[s = \\frac{t^3}{16}\\]
        \n\uc774\ub2e4.\n<\/li>\n<\/ol>\n<\/div>\n


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