vè;k; 7 7.6.3. ∫( px + q) ge vpj A vkSj B bl izdkj pqurs gSa fd ⎡ d 2 ⎤ px + q =A (ax + bx + c) + B ⎢⎥ ⎣ dx ⎦ = A(2ax + b) + B nksuksa i{kksa esa x osQ xq.kkadksa vkSj vpj inksa dh rqyuk djus ij] gesa izkIr gksrk gSµ 2aA= p vkSj Ab + B = q bu lehdj.kksa dks gy djus ij] AvkSj B osQ eku izkIr gks tkrs gSaA bl izdkj] lekdy fuEu esa ifjofrZr gks tkrk gSµ A (2 ax + b) ax 2 + bx + c dx + B∫ ax 2 + bx + c dx ∫ = AI+BI ] tgk¡ 12 I= (2 ax + b) 1 ∫ ax2 + bx + c = t, jf[k,A rc] (2ax + b)dx = dt gSA 23 2 vr%] I= (ax 2 + bx + c) + C1 13 blh izdkj] I= 2 ∫ ikB~; iqLrd osQ i`"B 328 ij 7-6-2 esa ppkZ fd, x, lekdy lw=k dk iz;ksx djosQ Kkr fd;k tkrk gSA bl izdkj] ∫( px + q) ax 2 + bx + c dx dk eku varr% Kkr dj fy;k tkrk gSA mnkgj.k 25 ∫ x 1+ x − x 2 dx Kkr dhft,A gy mij n'kkZ, xbZ fof/ viukrs gq,] ge fy[krs gSaµ ⎡ d 2 ⎤ A1 + x − x + B x = ⎢( )⎥ ⎣ dx ⎦ = A (1 – 2x) + B nksuksa i{kksa esa] x osQ xq.kkadksa vkSj vpj inksa dks cjkcj djus ij] gesa –2A = 1 vkSj A + B = 0 izkIr gksrk gSA bu lehdj.kksa dks gy djus ij] ge A = − 1 vkSj B = 1 izkIr djrs gSaA bl izdkj] 22 lekdy fuEu esa ijkofrZr gks tkrk gSµ 21 21 2 x 1 + x − x dx = −∫(1 − 2) x 1 + x − x dx + 1 + x − x dx ∫ ∫ 22 11 = − I + I (1) 21 2 2 2 I= (1 − 2) x 1 + x − x dx ij fopkj dhft,A 1 ∫ 1 + x – x 2 = t jf[k,A rc, (1 – 2x)dx = dt gSA 13 22 + 2) bl izdkj] I = (1 − x 1 + x − x dx = t dt = t C 1∫ 2 ∫ 23 1 2 23 2 1 = (1+ x − x )+ C , tgk¡ C1 dksbZ vpj gSA 3 vkxs] I2 = ∫ 1 + x − x 2 dx ij fopkj dhft,A 5 ⎛ 1⎞ 2 ;g lekdy = −⎜ x −⎟ dx ∫ 4 ⎝ 2 1 x −=t jf[k,A rc] dx = dt gSA 2 ⎛ 5 ⎞ 22vr%] I2 = ∫⎜ ⎟− t dt ⎝ 2 ⎠ iwjd ikB~; lkexzh 631 1 12t 5 25 −1 t − t +⋅ sin + C2 = 2424 5 1 (2x −1) 5 15 ⎛ 2x −1⎞ 2 −1 − (x − ) + sin + C = ⎜ ⎟ 2 224 28 ⎝ 5 ⎠ 5 ⎛ 2x − 1⎞ 2 −1 1 (2 x −1) 1 + x − x + sin ⎜ + C = ⎟ 2, ⎠ 48 ⎝ 5 tgk¡C2 dksbZ vpj gSA (1) esaI1 vkSj I2 osQ eku j[kus ij] gesa izkIr gksrk gSµ 31 2 21 2 2 x 1 + x − x dx = − (1 + x − x ) + (2 x − 1) 1 + x − x ∫ 38 5 −1 ⎛ 2x −1⎞ + sin + C, ⎜ ⎟ tgk¡ 16 ⎝ 5 ⎠ C1 + C2 C= − ,d vU; vpj gSA 2 iz'ukoyh 7.7 osQ var esa] fuEufyf[kr iz'u lfEefyr dhft, 12. x x + x 2 13. (x + 1) 2x 2 + 3 14. (x + 3) 3 − 4x − x 2 mÙkj 1 2 32(2 x +1) x 2 + x 11 2 12. ( x + x) −+ log | x ++ x + x | +C 3 8 16 2 12 3 x 2 3 2 23 (2 x + 3) 2 + 2x + 3 + log x + x + + C 13. 62 4 2 1 237 −1 ⎛ x + 2⎞ (x + 2) 3 − 4x − x 2 14. − (3 − 4 x − x ) + sin 2 ++ C ⎜ ⎟ 32 ⎝ 7 ⎠ 2 632 xf.kr vè;k; 10 10.7 vfn'k f=kd xq.kuiQy eku yhft, fd ,a bbb vkSj cb dksbZ rhu lfn'k gSaA ab vkSj (bb )× cb osQ vfn'k xq.kuiQy] vFkkZr~ b b b b bb bb a ⋅(b × c) dks abvkSj cb dk blh ozQe esa vfn'k f=kd xq.kuiQy dgrs gSaA bls[,, c] (;k , ab b bb [ abc]) }kjk O;Dr fd;k tkrk gSA bl izdkj] gesa izkIr gSµ b b bbb b [,, c] = a ⋅ (b × c) ab isz{k.k b 1. D;ksafd (bb × c) ,d lfn'k gS] blfy, b b bb a ⋅(bb × c) ,d vfn'k jkf'k gS] vFkkZr~ [ abb , c] , ,d vfn'k jkf'k gSA 2. T;kferh; :i ls] vfn'k f=kd xq.kuiQy dk eku rhu lfn'kksa b,b vkSj cb ls iznf'kZr vklUu ab Hkqtkvksa ls cus lekarj "kV~iQyd dk vk;ru gksrk gS (nsf[k, vko`Qfr 10-28)A fulansg] lekarj "kV~iQyd osQ vk/kj dks cukus b b vko`Qfr 10.28 okys lekarj prqHkZt dk {ks=kiQy b × c gSA bb vkSj cb dks varfoZ"V djus okys ry ij vfHkyac osQ vuqfn'kb izs{ksi gh bldh m¡pkbZ gS] bb tks b × c dh fn'kk esa ab dk ?kVd gSA vFkkZr~ ;g gSA vr%] lekarj "kV~iQyd dk vk;ru iwjd ikB~; lkexzh 633 = (b 2 c 3 – b 3 c 2) iˆ + (b 3 c 1 – b 1 c 3) ˆj + (b 1 c 2 – b 2 c 1 ) ˆk rFkk blhfy, b .( )a b ×c b b b b b = 1 2 3 3 2 2 3 1 1 3 3 1 2 2 1( – ) ( – ) ( – )a b c b c a b c b c a b c b c+ + aa a bb b 123 123 = ccc 123 b bb 4. ;fn ,vkSj c dksbZ rhu lfn'k gSa] rks ab bb b b bb bb b [ , ]=[,] = [ ,] ab,c bc, ac, ab (rhuksa lfn'kksa osQ pozQh; ozQep; ls vfn'k f=kd xq.kuiQy osQ eku esa dksbZ ifjorZu ugha gksrk gSA) b b b b b b eku yhft, fd a = aiˆ + a ˆj+ akˆ, b =biˆ + bjˆ + bkˆ rFkk c = ciˆ+ c ˆjkˆ gSA 1 23 123 123 rc] osQoy ns[kdj gh] gesa izkIr gksrk gSµ aa a 123 b bb bb b 123 [,,]= abc ccc 123 = a (bc – bc ) + a (bc – bc ) + a (bc – bc ) 1233223113 31221 = b (ac – ac ) + b (ac – ac ) + b (ac – ac ) 1322321331321 12 1b 2b 3b = 1c 2c 3c 1a 2a 3a b bb =[,,] bca bb b bb b blh izdkj] ikBd bldh tk¡p dj ldrs gSa fd[,, c ] = [ cab,] gSA ab , bb b bb bbbb vr%] [,, c ] = [ ,,] = [ cab,,] gSA ab bca b bb 5. vfn'k f=kd xq.kuiQy a .(b × c) esa] MkV (dot) vkSj ozQkWl (cross) dks ijLij cnyk tk ldrk gSA fuLlansg] bbb bbb bbbb bbbb bb b b .(b × c)= [abc ] =[ bc, a ] = [ cab,,] = c.( a × b)= ( a ×b ).c a ,,, bb b b bb 6. = [,,] = – [ a , ]. fuLlansg] abc , cb bb bbb b =[ ,,] = a .(b × c) abc b b b = a.(– c ×b) b bb = –(a .(c ×b )) b bb =– ⎡a, cb ⎤  , ⎦ b b b b b b ab fuLlansg] 7. [a,,]= 0. b b b b b b b b b b b b [aab, ,] [,, = aba ,] b b bb b bb b =[ , , baa] b b b b b b = b .(a ×a) b b b b b b b b b b = b .0 = 0. (D;ksafd a × a = 0) fVIi.kh mi;qZDr 7 esa] fn;k ifj.kke] nksuksa cjkcj lfn'kksa osQ fLFkfr;ksa osQ fdlh Hkh ozQe esa gksus ij Hkh lR; gSA 10-7-1 rhu lfn'kksa dh leryh;rk b b bb bb isze;1 rhu lfn'k a , b vkSj c leryh; gksrs gSa] ;fn vkSj osQoy ;fn a⋅ (b × c) = 0 gksrk gSA b bb miifÙk loZizFke] eku yhft, fd a , b vkSj c leryh; gSaA b b b b iwjd ikB~; lkexzh 635 bb b b bbb b ;fn b vkSj c lekarj ugha gS] rks b lfn'k a ij yac gksxk] D;kasfd a vkSj c leryh; gSaA × c , b b bb vr%] a ⋅ (b × c) = 0 gSA b b bb b b foykser%] eku yhft, fd a⋅ (b × c) = 0 gSA ;fn a vkSj b × c esa ls nksuksa 'kwU;srj lfn'k gSa] b bb b b b rks ge fu"d"kZ fudkyrs gSa fd a vkSj b nks ykafcd lfn'k gSaA ijarq b × c nksuksa lfn'kksa b ×c b b b b vkSj c ij yac gSA vr%] a ] b vkSj c ,d lery esa fLFkr gksus pkfg,] vFkkZr~ ;s leryh; b bb gSaA;fn a = 0 gS] rks a fdUgha Hkh nks lfn'kksa] fo'ks"k :i ls b vkSj c b ] osQ leryh; gksxkA ;fn b b b bb bb (b × c) = 0 gS] rks b vkSj c lekarj lfn'k gksaxs rFkk blhfy, a , b vkSj c leryh; gksaxs] D;kasfd dksbZ Hkh nks lfn'k lnSo ,d lery esa gksrs gSa] tks muls fu/kZfjr gksrk gS] rFkk dksbZ lfn'k] tks bu nksuksa lfn'kksa esa ls fdlh ,d lekarj gksrk gS] Hkh blh lery esa fLFkr gksrk gSA fVIi.kh pkj fcanqvksa dh leryh;rk dh ppkZ] rhu lfn'kksa dh leryh;rk dk iz;ksx djrs gq,] A A b A b A A b AAAA b AAA Ab AAAA b dh tk ldrh gSA fuLlansg] pkj fcanq A, B, C vkSjD leryh; gksrs gSa] ;fn lfn'k AB, AC vkjS AD leryh; gksaA b b b b b b b b b b b b ˆˆ mnkgj.k26 a .(b ×c) Kkr dhft,] ;fna = 2iˆ + ˆj +3k, b = – iˆ + 2 j+ k vkSj c =3iˆ + jˆ +2k gSA 2 1 3 b gy gesa izkIr gSµ .( ) 1a b ×c = − b b b b b 2 1 –10.= 3 1 2 mnkgj.k 27 n'kkZb, fd lfn'k leryh; gSaA gy gesa izkIr gSµ vr%] izes; 1 osQ vuqlkj leryh; lfn'k gSaA λ dk eku Kkr dhft,A gy D;kasfda ,b vkSj cleryh; gSa] blfy, ⎡⎣a,b, c ⎤⎦= 0, 1 3 1 vFkkZr~] 2 1− 1 0. − = λ 7 3 ⇒ 1 (– 3 + 7) – 3 (6 + λ) + 1 ( 14 + λ) = 0 ⇒λ = 0. mnkgj.k 29 n'kkZb, fd fLFkfr lfn'kksa 4iˆ + 5ˆj + kˆ,−(ˆj + kˆ),3 iˆ +9ˆj + 4kˆ vkSj 4(– iˆ + ˆj + kˆ) okys ozQe'k% pkjks fcanq A, B, C vkSj D leryh; gSaA gy ge tkurs gSa fd pkj fcanq A, B, C vkSjD leryh; gksrs gSa] ;fn rhuksa lfn'k AB,AC vkjS ADleryh; gksrs gSa] vFkkZr~] vr%] A, B, C vkSj D leryh; gSaA mnkgj.k 30 fl¼ dhft, fd gy gesa izkIr gSµ iwjd ikB~; lkexzh 637 (D;kasfd c × c = 0 gSA) =2 ,, ⎤ (D;ksa? ) ⎡abc ⎣⎦ mnkgj.k 31 fl¼ fdft, fd gy gesa izkIr gSµ = ⎡ ,, ⎤+⎡abd, ⎤ abc , ⎣⎦⎣ ⎦ iz'ukoyh 10-5 (mÙkj 24) leryh; gSa] rks λ dk eku Kkr dhft,A (mÙkj λ = 15) (a) ;fnc1 = 1 vkSjc2 = 2 gS] rks c3 Kkr dhft,] ftlls leryh; gks tk,¡A (mÙkj c 3 = 2) (b) ;fn c = –1 vkSj c = 1 gS] rks n'kkZb, fd c dk dksbZ Hkh eku ,vkj c dks ab Sleryh; ugha cuk ldrk gSA 23 1 5. n'kkZb, fd fLFkfr lfn'kksa 4iˆ + 8ˆj+12kˆ,2ˆi+ 4ˆj+6kˆ,3iˆ +5ˆj+ 4kˆ vkSj5ˆi+ 8ˆj+ 5kˆ okys pkjksa fcanq leryh; gSaA 6. ;fn pkj fcanq A (3, 2, 1), B (4, x, 5), C (4, 2, –2) vkSj D (6, 5, –1) leryh; gSa] rks x dk eku Kkr dhft,A (mÙkj x = 5) 7. ;fn a + bb, + c vkSjc + a leryh; gSa] rks n'kkZb, fd lfn'k ,jSc leryh; abvk gksaxsA