{"id":11152,"date":"2021-02-25T13:17:51","date_gmt":"2021-02-25T07:47:51","guid":{"rendered":"https:\/\/learnsteer.sasnaka.org\/science\/?p=11152"},"modified":"2021-10-12T23:22:29","modified_gmt":"2021-10-12T17:52:29","slug":"04-05-02-2","status":"publish","type":"post","link":"https:\/\/learnsteer.sasnaka.org\/science\/advanced-level-science\/combined-mathematics\/04-05-02-2\/","title":{"rendered":"04.05.02 – \u0dc3\u0db8\u0dca\u0db8\u0dad \u0d86\u0d9a\u0dcf\u0dbb – 1"},"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

\u0d9c\u0dd0\u0da7\u0dc5\u0dd4 \u0dba\u0dd9\u0daf\u0dd9\u0db1 \u0dc3\u0db8\u0dca\u0db8\u0dad \u0d86\u0d9a\u0dcf\u0dbb<\/h2>\r\n\r\n\r\n\r\n


1.\\int\\frac{px+q}{ax+b}dx\\;<\/span> \u0d86\u0d9a\u0dcf\u0dbb\u0dba<\/span><\/p>\r\n\r\n\r\n\r\n

px+q\\equiv\\frac pa\\left(ax+b\\right)+q-p\\frac ba<\/span><\/p>\r\n\r\n\r\n\r\n

\\begin{array}{rcl}\\int\\frac{px+q}{ax+b}dx&=&\\int\\frac{\\frac pa{\\displaystyle\\left(ax+b\\right)}{\\displaystyle+}{\\displaystyle q}{\\displaystyle-}{\\displaystyle p}\\frac ba}{ax+b}dx\\\\&=&\\int\\left(\\frac pa+\\frac{q-p\\frac ba}{ax+b}\\right)dx\\\\&=&\\frac pa\\int dx+\\left(q-p\\frac ba\\right)\\int\\frac1{ax+b}dx\\\\&=&\\frac pax+\\frac1a\\left(q-\\frac ba\\ln\\left|ax+b\\right|+c\\right)\\end{array}<\/span>:c \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba \u0dc0\u0dda
\u0d8b\u0daf\u0dcf:
1.\\int\\frac{2x+3}{3x-2}dx\\;<\/span><\/p>\r\n\r\n\r\n\r\n

\\begin{array}{rcl}2x+3&\\equiv&\\frac23\\left(3x-2\\right)+3-2\\frac{\\left(-2\\right)}3\\\\&\\equiv&\\frac23\\left(3x-2\\right)+\\frac{13}3\\end{array}<\/span><\/p>\r\n\r\n\r\n\r\n

\\begin{array}{rcl}\\int\\frac{2x+3}{3x-2}dx&=&\\int\\frac{\\frac23{\\displaystyle\\left(3x-2\\right)}{\\displaystyle+}\\frac{13}3}{3x-2}dx\\\\&=&\\int\\left(\\frac23+\\frac{\\frac{13}3}{3x-2}\\right)dx\\\\&=&\\frac23\\int dx+\\frac{13}3\\int\\frac1{3x-2}dx\\\\&=&\\frac23x+\\frac{13}{3.3}\\ln\\left|3x-2\\right|+c\\\\&=&\\frac23x+\\frac{13}9\\ln\\left|3x-2\\right|+c\\end{array}<\/span>:c \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba \u0dc0\u0dda
2.\\int\\frac{x+3}{x-2}dx\\;<\/span><\/p>\r\n\r\n\r\n\r\n

x+3\u2261x-2+5<\/span><\/p>\r\n\r\n\r\n\r\n

\\begin{array}{rcl}\\int\\frac{x+3}{x-2}dx&=&\\int\\frac{\\left(x-2\\right){\\displaystyle+}{\\displaystyle5}}{x-2}dx\\\\&=&\\int\\left(1+\\frac5{x-2}\\right)dx\\\\&=&\\int dx+5\\int\\frac1{x-2}dx\\\\&=&x+5\\ln\\left|x-2\\right|+c\\end{array}<\/span>:c \u0d85\u0db7\u0dd2\u0db8\u0dad \u0db1\u0dd2\u0dba\u0dad\u0dba \u0dc0\u0dda<\/p>\r\n\r\n\r\n\r\n

\u0dc4\u0dbb\u0dba\u0dda \u0d92\u0d9a\u0da2 \u0db4\u0dca\u200d\u0dbb\u0d9a\u0dcf\u0dc1\u0dba\u0d9a\u0dca \u0d87\u0dad\u0dd2 \u0dc0\u0dd2\u0da7 \u0dbd\u0dc0\u0dba\u0dda \u0d9a\u0dd4\u0db8\u0db1 \u0db8\u0dcf\u0dad\u0dca\u200d\u0dbb\u0dba\u0dda \u0db6\u0dc4\u0dd4\u0db4\u0daf \u0db4\u0dca\u200d\u0dbb\u0d9a\u0dcf\u0dc1\u0dba\u0d9a\u0dca \u0dad\u0dd2\u0db6\u0dd4\u0dab\u0daf \u0d9c\u0dd0\u0da7\u0dc5\u0dd4\u0dc0 \u0dc0\u0dd2\u0dc3\u0db3\u0db1\u0dca\u0db1\u0dda \u0db8\u0dda \u0d86\u0d9a\u0dcf\u0dbb\u0dba\u0da7\u0db8 \u0dc0\u0dda<\/p>\r\n\r\n\r\n\r\n

\u0d8b\u0daf\u0dcf:
1.\\int\\frac{x^3}{x-1}dx\\;<\/span>
\u0db8\u0dd9\u0dc4\u0dd2\u0daf\u0dd3 x3<\/sup>, (x-1) \u0db1\u0dca \u0db6\u0dd9\u0daf\u0dd3\u0db8\u0dd9\u0db1\u0dca \u0dc4\u0ddd \u0db6\u0dd9\u0daf\u0dd4\u0db8\u0dca \u0d87\u0dbd\u0dca\u0d9c\u0ddc\u0dbb\u0dd2\u0dad\u0db8\u0dba \u0dba\u0ddc\u0daf\u0dcf\u0d9c\u0dd0\u0db1\u0dd3\u0db8\u0dd9\u0db1\u0dca,<\/p>\r\n\r\n\r\n\r\n

x^3\\equiv\\left(x-1\\right)(x^2+x+1)+1<\/span><\/p>\r\n\r\n\r\n\r\n\\begin{array}{rcl}\\int\\frac{x^3}{x-1}dx&=&\\int\\frac{(x-1)(x^2+x+1)+1}{x-1}dx\\\\&=&\\int\\left(x^2+x+1+\\frac1{x-1}\\right)dx\\\\&=&\\int x^2dx+\\int xdx+\\int dx+\\int\\frac1{x-1}dx\\\\&=&\\frac13x^3+\\frac12x^2+\\ln\\left|x-1\\right|+c\\end{array}<\/span>\r\n\r\n\r\n\r\n

2.\\int\\frac{dx}{ax^2+bx+c}\\;<\/span> \u0d86\u0d9a\u0dcf\u0dbb\u0dba<\/span><\/p>\r\n\r\n\r\n\r\n

\u0dc4\u0dbb\u0dba\u0dda \u0d87\u0dad\u0dd2 \u0dc0\u0dbb\u0dca\u0d9c\u0da2 \u0db4\u0dca\u200d\u0dbb\u0d9a\u0dcf\u0dc1\u0db1\u0dba \u0dc3\u0dcf\u0db0\u0d9a \u0dc0\u0dbd\u0da7 \u0dc0\u0dd9\u0db1\u0dca \u0dc0\u0dd9\u0db1 \u0d85\u0dc0\u0dc3\u0dca\u0dae\u0dcf\u0dc0<\/span><\/h4>\r\n\r\n\r\n\r\n