{"id":11778,"date":"2021-05-19T16:36:50","date_gmt":"2021-05-19T11:06:50","guid":{"rendered":"https:\/\/learnsteer.sasnaka.org\/science\/?p=11778"},"modified":"2021-10-12T23:34:34","modified_gmt":"2021-10-12T18:04:34","slug":"04-05-07","status":"publish","type":"post","link":"https:\/\/learnsteer.sasnaka.org\/science\/advanced-level-science\/combined-mathematics\/04-05-07\/","title":{"rendered":"04.05.07 – \u0dc3\u0db8\u0dca\u0db8\u0dad \u0d86\u0daf\u0dda\u0dc1-2"},"content":{"rendered":"\r\n
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\u0db4\u0dcf\u0da9\u0db8\u0dda \u0dc3\u0da7\u0dc4\u0db1 Download \u0d9a\u0dbb\u0d9c\u0db1\u0dca\u0db1.<\/a><\/div>\r\n<\/div>\r\n\r\n\r\n\r\n

9.\\;\\int\\sqrt{\\mathbf a^{\\mathbf2}+\\mathbf x^{\\mathbf2}\\;}\\mathbf{dx}<\/span>\u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dda \u0d85\u0db1\u0dd4\u0d9a\u0dbd<\/span><\/strong><\/p>\r\n\r\n\r\n\r\n

    \r\n
  • \u0db8\u0dd9\u0dc0\u0dd0\u0db1\u0dd2 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1 x=a\\tan{\\theta}<\/span> \u0d86\u0daf\u0dda\u0dc1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dcf \u0d9a\u0dbb\u0db8\u0dd4.<\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n

    \u0d8b\u0daf\u0dcf.\\int\\sqrt{9+x^2}dx<\/span> \u0dc3\u0dbd\u0d9a\u0db8\u0dd4.
    x=3\\tan\\theta<\/span> \u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dbb\u0db8\u0dd4.
    x \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0db1\u0dca,
    dx=3\\sec^2\\theta\\;d\\theta<\/span>
    \\begin{array}{rcl}\\tan\\theta&=&\\frac x3\\\\\\sec\\theta&=&\\;\\sqrt{1+\\tan^2\\theta}\\\\\\sec\\theta&=&\\;\\sqrt{1+\\frac{x^2}9}=\\frac13\\sqrt{9+x^2}\\\\\\sec\\theta&=&\\frac13\\sqrt{9+x^2}\\end{array}<\/span>
    \\begin{array}{rcl}I&=&\\int\\sqrt{9+x^2}dx\\\\I&=&\\int\\sqrt{9+9\\tan^2\\theta}.3\\sec^2\\theta\\;d\\theta\\\\I&=&\\int\\sqrt{9(1+\\tan^2\\theta)}.3\\sec^2\\theta\\;d\\theta\\\\I&=&\\int\\sqrt{9\\sec^2\\theta}.3\\sec^2\\theta\\;d\\theta\\\\I&=&\\int3\\sec\\theta.3\\sec^2\\theta\\;d\\theta\\\\I&=&\\int9\\sec^3\\theta\\;d\\theta\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n

      \r\n
    • \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca \u0db8\u0dd9\u0dad\u0dd0\u0db1\u0dca \u0dc3\u0dd2\u0da7 \u0d9c\u0dab\u0db1 \u0dc0\u0dd2\u0dc3\u0daf\u0db8\u0dd4.<\/span><\/li>\r\n<\/ul>\r\n\r\n\r\n\r\n

      \\begin{array}{rcl}u&=&sec\\theta\\\\\\frac{du}{d\\theta}&=&sec\\theta tan\\theta\\\\\\frac{dv}{d\\theta}&=&sec2\\theta\\\\\\int dv&=&\\int sec^2\\theta d\\theta\\\\v&=&tan\\theta\\end{array}<\/span>
      \\begin{array}{l}I=9\\int\\sec\\theta.\\sec^2\\theta\\;d\\theta\\\\I=9\\sec\\theta\\tan\\theta-9\\int\\sec\\theta.\\tan^2\\theta\\;d\\theta\\\\I=9\\sec\\theta\\tan\\theta-9\\int\\sec\\theta.\\left(\\sec^2\\theta-1\\right)\\;d\\theta\\\\I=9\\sec\\theta\\tan\\theta+9\\int\\sec\\theta d\\theta-9\\int\\sec^3\\theta\\;d\\theta;(I=9\\int\\sec^3\\theta\\;d\\theta)\\\\2I=9\\sec\\theta\\tan\\theta+9\\ln{\\vert\\sec\\theta+\\tan{\\theta\\vert}}+c\\\\I=\\frac92.\\;\\frac13\\sqrt{9+x^2}.\\frac x3+\\frac92.\\ln{\\vert\\;}\\frac13\\sqrt{9+x^2}+\\frac x3\\vert+c\\\\I=\\frac32.x\\sqrt{9+x^2}.+\\frac92.\\ln{\\vert\\;}\\frac13\\sqrt{9+x^2}+\\frac x3\\vert+c\\text{ : c \u0dba\u0db1\u0dd4 \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2.}\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n\r\n\r\n

      10.\\;\\int\\sqrt{\\mathbf x^{\\mathbf2}-\\mathbf a^{\\mathbf2}\\;}\\mathbf{dx}<\/span>\u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dda \u0d85\u0db1\u0dd4\u0d9a\u0dbd<\/strong>
      <\/span>\u0db8\u0dd9\u0dc0\u0dd0\u0db1\u0dd2 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1 x=a\\sec\\theta<\/span> \u0d86\u0daf\u0dda\u0dc1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dcf \u0d9a\u0dbb\u0db8\u0dd4.
      \u0d8b\u0daf\u0dcf.
      \\int\\sqrt{x^2-7}dx<\/span> \u0dc3\u0dbd\u0d9a\u0db8\u0dd4.
      x=\\sqrt{7\\;}\\sec\\theta<\/span> \u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dbb\u0db8\u0dd4.
      x \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0db1\u0dca,
      dx=\\sqrt{7\\;}\\sec\\theta\\;tan\\theta\\;d\\theta<\/span>
      \\begin{array}{rcl}\\sec\\theta&=&\\frac x{\\sqrt7}\\\\\\tan\\theta&=&\\sqrt{\\sec^2\\theta-1}\\\\\\tan\\theta&=&\\sqrt{\\frac{x^2}7-1}J=\\int\\sqrt{x^2-7}dx\\\\J&=&\\int\\sqrt{7\\sec^2\\theta-7}.\\sqrt7\\sec\\theta\\;tan\\theta\\;d\\theta\\\\J&=&\\int\\sqrt{{7(\\sec}^2\\theta-1)}.\\sqrt7\\sec\\theta\\;tan\\theta\\;d\\theta\\\\J&=&\\int\\sqrt{7\\tan^2\\theta}.\\sqrt7\\sec\\theta\\;tan\\theta\\;d\\theta\\\\J&=&\\int\\sqrt7\\tan\\theta.\\sqrt7\\sec\\theta\\;tan\\theta\\;d\\theta\\\\J&=&\\int7\\sec\\theta\\;tan\\;^2\\theta\\;d\\theta\\\\J&=&7\\int\\sec\\theta\\;{(\\sec}^2\\theta-1)\\;d\\theta\\\\J&=&7\\int\\sec^3\\theta\\;d\\theta-7\\int\\sec{\\theta\\;d\\theta\\rightarrow(1)}\\end{array}<\/span>
      J_1=7\\int\\sec^3\\theta\\;d\\theta<\/span> \u0dbd\u0dd9\u0dc3 \u0d9c\u0db1\u0dd2\u0db8\u0dd4.J_1=7\\int\\sec\\theta\\sec^2\\theta\\;d\\theta<\/span>
      \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca \u0db8\u0dd9\u0dad\u0dd0\u0db1\u0dca \u0dc3\u0dd2\u0da7 \u0d9c\u0dab\u0db1 \u0dc0\u0dd2\u0dc3\u0daf\u0db8\u0dd4.
      <\/span>\\begin{array}{rcl}u&=&sec\\theta\\\\\\frac{du}{d\\theta}&=&sec\\theta tan\\theta\\\\\\frac{dv}{d\\theta}&=&sec^2\\theta\\\\\\frac{dv}{d\\theta}&=&sec^2\\theta d\\theta\\\\v&=&tan\\theta\\end{array}<\/span>
      \\begin{array}{rcl}J_1&=&7\\int\\sec\\theta.\\sec^2\\theta\\;d\\theta\\\\J_1&=&7\\sec\\theta\\tan\\theta-7\\int\\sec\\theta.\\tan^2\\theta\\;d\\theta\\\\J_1&=&7\\sec\\theta\\tan\\theta-7\\int\\sec\\theta.\\left(\\sec^2\\theta-1\\right)\\;d\\theta\\\\J_1&=&7\\sec\\theta\\tan\\theta+7\\int\\sec\\theta d\\theta-7\\int\\sec^3\\theta\\;d\\theta;(J_1=7\\int\\sec^3\\theta\\;d\\theta)\\\\2J_1&=&7\\sec\\theta\\tan\\theta+7\\int\\sec\\theta d\\theta\\\\J_1&=&\\frac72\\sec\\theta\\tan\\theta+\\frac72\\int\\sec\\theta d\\theta\\end{array}<\/span>
      \u0daf\u0dd0\u0db1\u0dca J_1=7\\int\\sec^3\\theta\\;d\\theta<\/span> ,(1) \u0dc4\u0dd2 \u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dbb\u0db8\u0dd4.
      <\/span>\\begin{array}{rcl}J&=&\\frac72\\sec\\theta\\tan\\theta+\\frac72\\int\\sec\\theta d\\theta-7\\int\\sec\\theta\\;d\\theta\\\\J&=&\\frac72\\sec\\theta\\tan\\theta-\\frac72\\int\\sec\\theta d\\theta\\\\J&=&\\frac72\\sec\\theta\\tan\\theta-\\frac72\\ln{\\vert\\sec\\theta+\\tan{\\theta\\vert}}+c\\\\J&=&\\frac72.\\;\\frac x{\\sqrt7}.\\;\\sqrt{\\frac{x^2}7-1}-\\frac72.\\ln\\left|\\frac x{\\sqrt7}+\\sqrt{\\frac{x^2}7-1}\\right|+c\\\\J&=&\\;\\frac x2.\\;\\sqrt{x^2-7}-\\frac72.\\ln\\left|\\frac x{\\sqrt7}+\\sqrt{\\frac{x^2}7-1}\\right|+c\\end{array}<\/span>:c \u0dba\u0db1\u0dd4 \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n

      11.\\;\\int\\left(\\mathbf a^{\\mathbf2}-\\mathbf x^{\\mathbf2}\\right)^{\\mathbf3\/\\mathbf2}<\/span>\u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dda \u0d85\u0db1\u0dd4\u0d9a\u0dbd<\/span><\/strong><\/p>\r\n\r\n\r\n\r\n

        \r\n
      • \u0db8\u0dd9\u0dc0\u0dd0\u0db1\u0dd2 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1x=a\\sin\\theta<\/span> \u0dc4\u0ddd x=a\\cos\\theta<\/span> \u0d86\u0daf\u0dda\u0dc1 \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dcf \u0d9a\u0dbb\u0db8\u0dd4.
        \u0d8b\u0daf\u0dcf.<\/li>\r\n<\/ul>\r\n

        \\int\\left(4-x^2\\right)^{3\/2}dx<\/span>\u0dc3\u0dbd\u0d9a\u0db8\u0dd4.
        x=2\\sin\\theta<\/span> \u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dbb\u0db8\u0dd4.
        x \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0db1\u0dca,
        \\begin{array}{rcl}dx&=&2\\;cos\\theta\\;d\\theta\\\\\\cos\\theta&=&\\;\\sqrt{1-\\sin^2\\theta}=\\;\\sqrt{1-\\frac{x^2}4}=\\;\\frac{\\sqrt{4-x^2}}2\\end{array}<\/span>\\begin{array}{l}I=\\int\\left(4-x^2\\right)^{3\/2}dx\\\\I=\\int\\sqrt{4-x^2}\\left(4-x^2\\right)dx\\\\I=\\int\\sqrt{4-4\\sin^2\\theta}.\\left(4-4\\sin^2\\theta\\right).2\\cos\\theta d\\theta\\\\I=\\int\\sqrt{4\\cos^2\\theta}.\\left(4\\cos^2\\theta\\right).2\\cos\\theta d\\theta=16\\int\\cos^4\\theta\\;d\\theta\\\\I=16\\int\\cos^3\\theta.\\cos\\theta\\;d\\theta\\end{array}<\/span>
        \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca \u0db8\u0dd9\u0dad\u0dd0\u0db1\u0dca \u0dc3\u0dd2\u0da7 \u0d9c\u0dab\u0db1 \u0dc0\u0dd2\u0dc3\u0daf\u0db8\u0dd4.
        <\/span>\\begin{array}{rcl}u&=&\\cos^3\\theta\\\\\\frac{du}{d\\theta}&=&-3\\cos^3\\theta\\sin\\theta\\\\\\frac{dv}{d\\theta}&=&\\cos\\theta\\\\\\int dv&=&\\int\\cos\\theta\\;d\\theta\\\\v&=&\\sin\\theta\\end{array}<\/span>
        \\begin{array}{l}I=16\\int\\cos^3\\theta.\\cos\\theta\\;d\\theta\\\\I=16\\sin\\theta\\cos^3\\theta-16\\int\\sin\\theta.{(-3\\cos}^2\\theta.\\sin{\\theta)}\\;d\\theta\\\\I=16\\sin\\theta\\cos^3\\theta+48\\int\\sin^2\\theta.\\cos^2\\theta.\\;d\\theta\\\\I=16\\sin\\theta\\cos^3\\theta+48\\int{{(1-\\cos}^2\\theta).\\cos^2\\theta.\\;d\\theta}\\\\I=16\\sin\\theta\\cos^3\\theta+48\\int\\cos^2\\theta.\\;d\\theta-48\\int\\cos^4\\theta d\\theta\\\\I=16\\sin\\theta\\cos^3\\theta+48\\int{\\cos^2\\theta.\\;d\\theta-3I(\\;;}\\;I=16cos4\\theta d\\theta\\text{\u0db6\u0dd0\u0dc0\u0dd2\u0db1\u0dca})\\\\4I=16\\sin\\theta\\cos^3\\theta+48\\int\\frac{\\left(1+\\cos2\\theta\\right)}2d\\theta\\\\I=4\\sin\\theta\\cos^3\\theta+12\\int d\\theta+12\\int\\cos2\\theta\\;d\\theta\\\\I=4\\sin\\theta\\cos^3\\theta+12\\;\\theta+12.\\frac{\\sin2\\theta}2+c\\\\I=4.\\frac x2.\\left(1-\\frac{x^2}4\\right)^{3\/2}+12\\;\\sin^{-1}\\left(\\frac x2\\right)+6\\sin\\left[2\\;\\left(\\sin^{-1}\\frac x2\\right)\\right]+c\\\\I=2x\\left(1-\\frac{x^2}4\\right)^{3\/2}+12\\;\\sin^{-1}\\left(\\frac x2\\right)+6\\sin\\left[2\\;\\left(\\sin^{-1}\\frac x2\\right)\\right]+c\\text{ : c \u0dba\u0db1\u0dd4 \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2.}\\end{array}<\/span><\/p>\r\n


        12.\\int\\left(\\mathbf a^{\\mathbf2}+\\mathbf x^{\\mathbf2}\\right)^{\\mathbf3\/\\mathbf2}dx<\/span>\u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dda \u0d85\u0db1\u0dd4\u0d9a\u0dbd<\/span><\/strong><\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n

          \r\n
        • \u0db8\u0dd9\u0dc0\u0dd0\u0db1\u0dd2 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1 x=a\\tan\\theta<\/span>\u0d86\u0daf\u0dda\u0dc1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dcf \u0d9a\u0dbb\u0db8\u0dd4.
          \u0d8b\u0daf\u0dcf.<\/li>\r\n<\/ul>\r\n

          \\int\\left(1+x^2\\right)^{3\/2}dx<\/span>\u0dc3\u0dbd\u0d9a\u0db8\u0dd4.
          x=\\tan\\theta<\/span> \u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dbb\u0db8\u0dd4.
          \\sec\\theta=\\sqrt{1+x^2}<\/span>
          x \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0db1\u0dca,
          dx=\\sec^2\\theta\\;d\\theta<\/span>
          \\begin{array}{rcl}I&=&\\int\\left(1+x^2\\right)^{3\/2}dx\\\\I&=&\\int\\sqrt{1+x^2}\\left(1+x^2\\right)dx\\\\I&=&\\int\\sqrt{1+\\tan^2\\theta}.\\left(1+\\tan^2\\theta\\right).\\sec^2\\theta d\\theta\\\\I&=&\\int\\sqrt{\\sec^2\\theta}.\\left(\\sec^2\\theta\\right).\\sec^2\\theta d\\theta\\\\I&=&\\int\\sec^5\\theta\\;d\\theta\\\\I&=&\\int\\sec^3\\theta.\\sec^2\\theta\\;d\\theta\\end{array}<\/span>
          \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca \u0db8\u0dd9\u0dad\u0dd0\u0db1\u0dca \u0dc3\u0dd2\u0da7 \u0d9c\u0dab\u0db1 \u0dc0\u0dd2\u0dc3\u0daf\u0db8\u0dd4.
          <\/span>\\begin{array}{rcl}u&=&\\sin^3\\theta\\\\\\frac{du}{d\\theta}&=&3sec^2\\theta sec\\theta\\tan\\theta\\\\\\frac{du}{d\\theta}&=&3\\tan\\theta sec^3d\\theta\\\\\\frac{dv}{d\\theta}&=&sec^2\\theta\\\\\\int dv&=&\\int sec^2\\theta d\\theta\\\\v&=&\\tan\\theta\\\\&&\\\\&&\\\\&&\\end{array}<\/span>
          \\begin{array}{rcl}I&=&\\int\\sec^3\\theta.\\sec^2\\theta\\;d\\theta\\\\I&=&tan\\;\\theta\\sec^3\\theta-\\int\\tan\\theta.{(3\\sec}^3\\theta.\\tan{\\theta)}\\;d\\theta\\\\I&=&tan\\;\\theta\\sec^3\\theta-3\\int\\sec^3\\theta.\\tan^2\\theta.\\;d\\theta\\\\I&=&tan\\;\\theta\\sec^3\\theta-3\\int{\\sec^3\\theta.{(\\sec}^2\\theta-1).\\;d\\theta}\\\\I&=&tan\\;\\theta\\sec^3\\theta-3\\int\\sec^5\\theta.\\;d\\theta+3\\int\\sec^3\\theta d\\theta\\\\I&=&tan\\;\\theta\\sec^3\\theta-3I+3\\int\\sec^3\\theta d\\theta\\;(;I=sec5\\theta d\\theta\\text{\u0db6\u0dd0\u0dc0\u0dd2\u0db1\u0dca)}\\\\4I&=&tan\\;\\theta\\sec^3\\theta+3\\int\\sec^3\\theta d\\theta\\;\\rightarrow(1)\\end{array}<\/span>
          J_1=3\\int\\sec^3\\theta\\;d\\theta<\/span> \u0dbd\u0dd9\u0dc3 \u0d9c\u0db1\u0dd2\u0db8\u0dd4.J_1=3\\int\\sec\\theta\\sec^2\\theta\\;d\\theta<\/span>
          \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca \u0db8\u0dd9\u0dad\u0dd0\u0db1\u0dca \u0dc3\u0dd2\u0da7 \u0d9c\u0dab\u0db1 \u0dc0\u0dd2\u0dc3\u0daf\u0db8\u0dd4.
          <\/span>\\begin{array}{rcl}u&=&sec\\theta\\\\\\frac{du}{d\\theta}&=&sec\\theta.\\tan\\theta\\\\\\frac{dv}{d\\theta}&=&sec^2\\theta\\\\\\int dv&=&\\int sec^2\\theta d\\theta\\\\v&=&\\tan\\theta\\\\&&\\\\&&\\end{array}<\/span>
          \\begin{array}{rcl}J_1&=&3\\int\\sec\\theta.\\sec^2\\theta\\;d\\theta\\\\J_1&=&3\\sec\\theta\\tan\\theta-3\\int\\sec\\theta.\\tan^2\\theta\\;d\\theta\\\\J_1&=&3\\sec\\theta\\tan\\theta-3\\int\\sec\\theta.\\left(\\sec^2\\theta-1\\right)\\;d\\theta\\\\J_1&=&3\\sec\\theta\\tan\\theta+3\\int\\sec\\theta d\\theta-3\\int\\sec^3\\theta\\;d\\theta;(J_1=3sec3\\theta d\\theta\\text{\u0db6\u0dd0\u0dc0\u0dd2\u0db1\u0dca})\\\\2J_1&=&3\\sec\\theta\\tan\\theta+3\\int\\sec\\theta d\\theta\\\\J_1&=&\\frac32\\sec\\theta\\tan\\theta+\\frac32\\int\\sec\\theta d\\theta\\\\J_1&=&3\\int\\sec^3\\theta\\;d\\theta,(1)\\text{\u0dc4\u0dd2\u0d86\u0daf\u0dda\u0dc1\u0d9a\u0dbb\u0db8\u0dd4.}\\end{array}<\/span>
          \\begin{array}{l}J_1=3\\int\\sec\\theta.\\sec^2\\theta\\;d\\theta\\\\J_1=3\\sec\\theta\\tan\\theta-3\\int\\sec\\theta.\\tan^2\\theta\\;d\\theta\\\\J_1=3\\sec\\theta\\tan\\theta-3\\int\\sec\\theta.\\left(\\sec^2\\theta-1\\right)\\;d\\theta\\\\J_1=3\\sec\\theta\\tan\\theta+3\\int\\sec\\theta d\\theta-3\\int\\sec^3\\theta\\;d\\theta;(J_1=3sec3\\theta d\\theta\\text{\u0db6\u0dd0\u0dc0\u0dd2\u0db1\u0dca})\\\\2J_1=3\\sec\\theta\\tan\\theta+3\\int\\sec\\theta d\\theta\\\\J_1=\\frac32\\sec\\theta\\tan\\theta+\\frac32\\int\\sec\\theta d\\theta\\text{ : c \u0dba\u0db1\u0dd4 \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2.}\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n

          13.\\;\\int\\left(\\mathbf x^{\\mathbf2}-\\mathbf a^{\\mathbf2}\\right)^{\\mathbf3\/\\mathbf2}dx<\/span> \u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0dda \u0d85\u0db1\u0dd4\u0d9a\u0dbd<\/span><\/strong><\/p>\r\n\r\n\r\n\r\n

            \r\n
          • \u0db8\u0dd9\u0dc0\u0dd0\u0db1\u0dd2 \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1 x=asec\\;\\theta<\/span> \u0d86\u0daf\u0dda\u0dc1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dcf \u0d9a\u0dbb\u0db8\u0dd4.
            \u0d8b\u0daf\u0dcf.<\/li>\r\n<\/ul>\r\n

            \\int\\left(x^2-1\\right)^{3\/2}dx\\;<\/span>\u0dc3\u0dbd\u0d9a\u0db8\u0dd4.
            x=\\sec\\theta<\/span>\u0d86\u0daf\u0dda\u0dc1 \u0d9a\u0dbb\u0db8\u0dd4.
            \\tan\\theta=\\sqrt{x^2-1}<\/span>
            x \u0dc0\u0dd2\u0dc2\u0dba\u0dd9\u0db1\u0dca \u0d85\u0dc0\u0d9a\u0dbd\u0db1\u0dba\u0dd9\u0db1\u0dca,
            dx=sec\\theta\\tan\\theta\\;d\\theta<\/span>
            \\begin{array}{rcl}I&=&\\int\\left(x^2-1\\right)^{3\/2}dx\\\\I&=&\\int\\sqrt{x^2-1}\\left(x^2-1\\right)dx\\\\I&=&\\int\\sqrt{\\sec^2\\theta-1}.\\left(\\sec^2\\theta-1\\right).\\sec\\theta\\tan\\theta d\\theta\\\\I&=&\\int\\sqrt{\\tan^2\\theta}.\\left(\\tan^2\\theta\\right).sec\\theta tan\\theta d\\theta\\\\I&=&\\int sec\\;\\theta\\tan^4\\theta\\;d\\theta I=\\int sec\\;\\theta\\;\\left(\\sec^2\\theta-1\\right)^2\\;d\\theta\\\\I&=&\\int sec\\theta\\;\\left(\\sec^4\\theta-2\\sec^2\\theta+1\\right)d\\theta\\;\\\\I&=&\\int\\sec^5\\theta d\\theta-2\\int\\sec^3\\theta d\\theta+\\int{sec\\theta d\\theta\\rightarrow(1)}\\end{array}<\/span>
            \\begin{array}{rcl}J&=&\\int\\sec^5\\theta\\;d\\theta\\text{\u0dbd\u0dd9\u0dc3\u0d9c\u0db1\u0dd2\u0db8\u0dd4}.\\\\J&=&\\int\\sec^3\\theta.\\sec^2\\theta\\;d\\theta\\end{array}<\/span>
            \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca \u0db8\u0dd9\u0dad\u0dd0\u0db1\u0dca \u0dc3\u0dd2\u0da7 \u0d9c\u0dab\u0db1 \u0dc0\u0dd2\u0dc3\u0daf\u0db8\u0dd4.
            <\/span>\\begin{array}{rcl}u&=&\\sin^3\\theta\\\\\\frac{du}{d\\theta}&=&3sec^2\\theta sec\\theta\\tan\\theta\\\\\\frac{du}{d\\theta}&=&3\\tan\\theta sec^3d\\theta\\\\\\frac{dv}{d\\theta}&=&sec^2\\theta\\\\\\int dv&=&\\int sec^2\\theta d\\theta\\\\v&=&\\tan\\theta\\\\&&\\\\&&\\\\&&\\end{array}<\/span>
            \\begin{array}{rcl}J&=&\\int\\sec^3\\theta.\\sec^2\\theta\\;d\\theta\\\\J&=&tan\\;\\theta\\sec^3\\theta-\\int\\tan\\theta.{(3\\sec}^3\\theta.\\tan{\\theta)}\\;d\\theta\\\\J&=&tan\\;\\theta\\sec^3\\theta-3\\int\\sec^3\\theta.\\tan^2\\theta.\\;d\\theta\\\\J&=&tan\\;\\theta\\sec^3\\theta-3\\int{\\sec^3\\theta.{(\\sec}^2\\theta-1).\\;d\\theta}\\\\J&=&tan\\;\\theta\\sec^3\\theta-3\\int\\sec^5\\theta.\\;d\\theta+3\\int\\sec^3\\theta d\\theta\\\\J&=&tan\\;\\theta\\sec^3\\theta-3I+3\\int\\sec^3\\theta d\\theta(;J=sec5\\theta d\\theta\\text{\u0db6\u0dd0\u0dc0\u0dd2\u0db1\u0dca})\\\\4J&=&tan\\;\\theta\\sec^3\\theta+3\\int\\sec^3\\theta d\\theta\\;\\\\J&=&\\frac14tan\\;\\theta\\sec^3\\theta+\\frac34\\int\\sec^3\\theta d\\theta\\rightarrow(2)\\end{array}<\/span>
            \\begin{array}{rcl}J_1&=&\\int\\sec^3\\theta\\;d\\theta\\text{\u00a0\u0dbd\u0dd9\u0dc3\u0d9c\u0db1\u0dd2\u0db8\u0dd4.}\\\\{\\text{J}}_1&=&\\int sec\\theta sec^2\\theta\\;d\\theta\\end{array}<\/span>
            \u0d9a\u0ddc\u0da7\u0dc3\u0dca \u0dc0\u0dc1\u0dba\u0dd9\u0db1\u0dca \u0d85\u0db1\u0dd4\u0d9a\u0dbd\u0db1\u0dba \u0db7\u0dcf\u0dc0\u0dd2\u0dad\u0dba\u0dd9\u0db1\u0dca \u0db8\u0dd9\u0dad\u0dd0\u0db1\u0dca \u0dc3\u0dd2\u0da7 \u0d9c\u0dab\u0db1 \u0dc0\u0dd2\u0dc3\u0daf\u0db8\u0dd4.
            <\/span>\\begin{array}{rcl}u&=&sec\\theta\\\\\\frac{du}{d\\theta}&=&sec\\theta.\\tan\\theta\\\\\\frac{dv}{d\\theta}&=&sec^2\\theta\\\\\\int dv&=&\\int sec^2\\theta d\\theta\\\\v&=&\\tan\\theta\\\\&&\\\\&&\\end{array}<\/span>
            \\begin{array}{rcl}J_1&=&\\int\\sec\\theta.\\sec^2\\theta\\;d\\theta J_1\\\\&=&\\sec\\theta\\tan\\theta-\\int\\sec\\theta.\\tan^2\\theta\\;d\\theta\\\\J_1&=&\\sec\\theta\\tan\\theta-\\int\\sec\\theta.\\left(\\sec^2\\theta-1\\right)\\;d\\theta\\\\J_1&=&\\sec\\theta\\tan\\theta+\\int\\sec\\theta d\\theta-\\int\\sec^3\\theta\\;d\\theta;(J_1=sec3\\theta d\\theta\\text{\u0db6\u0dd0\u0dc0\u0dd2\u0db1\u0dca})\\\\2J_1&=&\\sec\\theta\\tan\\theta+\\int\\sec\\theta d\\theta\\\\J_1&=&\\frac12\\sec\\theta\\tan\\theta+\\frac12\\int\\sec\\theta d\\theta\\rightarrow\\left(3\\right)\\\\&&\\text{(2)\u0db1\u0dca(1)\u0da7\u0d86\u0daf\u0dda\u0dc1\u0d9a\u0dbb\u0db8\u0dd4.}\\end{array}<\/span>
            \\begin{array}{rcl}I&=&\\frac14tan\\;\\theta\\sec^3\\theta+\\frac34\\int\\sec^3\\theta d\\theta-2\\int\\sec^3\\theta d\\theta+\\int\\sec\\theta d\\theta\\\\I&=&\\frac14tan\\;\\theta\\sec^3\\theta-\\frac54\\int\\sec^3\\theta d\\theta+\\int{sec\\theta d\\theta\\;\\rightarrow(4)}\\\\&&\\text{(3)\u0db1\u0dca(4)\u0da7\u0d86\u0daf\u0dda\u0dc1\u0d9a\u0dbb\u0db8\u0dd4.}\\end{array}<\/span>
            \\begin{array}{l}I=\\frac14sec^3\\theta\\;-\\;\\frac54\\left\\{\\frac12sec\\theta\\;\\tan\\theta\\;+\\frac12\\int sec\\theta\\;d\\theta\\right\\}+\\int sec\\theta\\;d\\theta\\\\I=\\frac14sec^3\\theta\\;-\\;\\frac58sec\\theta\\;\\tan\\theta\\;-\\frac58\\int sec\\theta\\;d\\theta+\\int sec\\theta\\;d\\theta\\\\I=\\frac14sec^3\\theta\\;-\\;\\frac58sec\\theta\\;\\tan\\theta\\;+\\frac38\\int sec\\theta\\;d\\theta\\\\I=\\frac14sec^3\\theta\\;-\\;\\frac58sec\\theta\\;\\tan\\theta\\;+\\frac38\\ln\\left|sec\\theta\\;+\\tan\\theta\\;\\right|+c\\\\I=\\frac14.\\sqrt{x^2-1}.x^3\\;-\\frac58.x.\\sqrt{x^2-1}\\;+\\frac38\\ln\\left|x+\\sqrt{x^2-1}\\right|\\;+c\\text{ : c \u0dba\u0db1\u0dd4 \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba\u0d9a\u0dd2.}\\end{array}<\/span><\/p>\r\n