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ЧЭНХБААТАР Өвөрхангай 2020 он

Ãàð÷èã 0.1 Èíòåãðàëûí áîäëîãî . . . . . . . . . . . . . . . . . . . . . . . . . . 2 0.2 0.1.1 Îíîëûí õàíäëàãà . . . . . . . . . . . . . . . . . . . . . . . . 2 0.3 0.4 0.1.2 Õÿëáàð áîäëîãóóä . . . . . . . . . . . . . . . . . . . . . . . . 2 0.5 Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä . . . . . . . . . . . . . . . . 8 0.2.1 Òîäîðõîéã³é èíòåãðàë . . . . . . . . . . . . . . . . . . . . . 8 0.2.2 Çàðèì ôóíêöèéí èíòåãðàë . . . . . . . . . . . . . . . . . . . 8 0.2.3 Òîäîðõîéã³é èíòåãðàëûí ÷àíàðóóä . . . . . . . . . . . . . . 9 0.2.4 Îðëóóëàõ àðãààð áîäîõ . . . . . . . . . . . . . . . . . . . . . 17 0.2.5 Õýñýã÷ëýí èíòåãðàë÷ëàõ . . . . . . . . . . . . . . . . . . . . 24 0.2.6 35 ax + bx + c2 õýëáýðèéí èíòåãðàë áîäîõ . . . . . . . . . . . . Ðàöèîíàë ôóíêöèéí èíòåãðàë . . . . . . . . . . . . . . . . . . . . . 44 0.3.1 Òîäîðõîé áóñ êîýôôèöèåíòèéí àðãààð çàäëàõ . . . . . . . . 44 0.3.2 Òîäîðõîé áóñ êîýôôèöèåíòèéí àðãà àøèãëàí èíòåãðàë áîäîõ 53 Èððàöèîíàë ôóíêöèéí èíòåãðàë . . . . . . . . . . . . . . . . . . . 70 70 0.4.1 n ax + b 80 ôóíêöèéí èíòåãðàë . . . . . . . . . . . . . . . . . cx + d 0.4.2 Weierstrass îðëóóëàõ . . . . . . . . . . . . . . . . . . . . . . Òðèãîíîìåòðèéí èíòåãðàë . . . . . . . . . . . . . . . . . . . . . . . 94 0.5.1 cos ax cos bxdx, sin ax cos bxdx, sin ax sin bxdx, sinm x cosn xdx, tg xdxn 94 èíòåãðàë . . . . . . . . . . . . 1

0.1. Èíòåãðàëûí áîäëîãî 2 0.1 Èíòåãðàëûí áîäëîãî 0.1.1 Îíîëûí õàíäëàãà Õýðâýý F (x) áà f (x) ôóíêö³³ä I çàâñàðò òîäîðõîéëîãäîæ áàéâàë F (x) = f (x), x ∈ I, òýãâýë f (x)-ûí ýõ ôóíêö íü F (x) áàéíà. Ýõ ôóíêö îëîõ íü äèôôåðåíöèàëûí óðâóó ³éëäýë þì. Æèøýý: x2 ýõ ôóíêö íü x3 áàéíà. Ó÷èð íü 3 x3 1 x3 = 1 · 3x2 = x2. = 33 3 x2 ýíý ôóíêöèéí ýõ ôóíêö íü x3 x3 − 2, x3 + C áàéíà. + 5, 3 33 x3 x3 + C = x2 + 0 = x2. +C = 33 f (x)-í ýõ ôóíêö íü F (x), I çàâñàðò òîäîðõîéëîãäñîí áîë f (x) ýõ ôóíêö íü F (x) + C áîëíî.C-òîãòìîë òîî 0.1.2 Õÿëáàð áîäëîãóóä f (x)-í ýõ ôóíêöèéã îëîõ íü äèôôåðåíöèàë òýãøèòãýëèéí øèéäèéã îëîõòîé àäèë. dy = f (x) y (x) = f (x) . dx Åð°íõèé øèéä íü äèôôåðåíöèàë òýãøèòãýë áà àíõíû í°õö- ëèéã õî¼óëàíã íü õàíãàäàã ôóíêö þì.Òóõàéí øèéäèéã îëî-

Ãàð÷èã 3 õûí òóëä àíõíû í°õöëèéã àøèãëàæ,-èéí óòãûã îëîõ ¼ñòîé. Æèøýý 1. 1 f (x) = x4 ôóíêöèéí ýõ ôóíêöèéã îë. 1 x3 ôóíêöèéí óëàìæëàëûã îëáîë 1 = x−3 = −3x−3−1 = −3x−4 = 3 x3 −x4 . Òèéìýýñ ýõ ôóíêö íü F (x) = −1 . 3x3 Øàëãàâàë 1 = −1 x−3 F (x) = − 3 3x3 = −1 · (−3) x−4 = x−4 = 1 = f (x) . 3 x4 Æèøýý 2. f (x) = e2x ôóíêöèéí ýõ ôóíêöèéã îë. Òîîöîîëîõîä õÿëáàð òîõèîëäîë e2x = e2x · (2x) = 2e2x. Ýõ ôóíêö íü e2x = 1 e2x = 1 · 2e2x = e2x = f (x) . F (x) = . 22 2 Øàëãàâàë e2x F (x) = 2

0.1. Èíòåãðàëûí áîäëîãî 4 Æèøýý 3. 1 f (x) = √ ôóíêöèéí ýõ ôóíêöèéã îë. 3x √ 2 = 2 x 2 −1 = 2 x− 1 = 2 = √2 . 3 x2 3 3 33x = x3 1 3 3 3x 3 Ýõ ôóíêö íü √ 3 3 x2 F (x) = 2 Øàëãàâàë 3 √ = 3 · √2 = √1 = f (x) . = 3 x2 √ 2 2 33x 3x 3 3 x2 F (x) = 2 Æèøýý 4. f (x) = 3−x ôóíêöèéí ýõ ôóíêöèéã îë. (3x) = 3x ln 3. Èéìýýñ 3−x = −3−x ln 3 áîëíî. Ýõ ôóíêöèéã îëáîë F (x) = − 1 · 3−x = −3−x . ln 3 ln 3 Øàëãàâàë = − 1 3−x = − 1 · −3−x ln 3 ln 3 ln 3 F (x) = −3−x ln 3 = 3−x. Æèøýý 5. dy = x2 − 1, y (3) = 7 òóõàéí øèéäèéã îë. dx

Ãàð÷èã 5 x2 − 1 ýõ ôóíêö íü y = x3 − x + C áîëíî. 3 C-èéí óòãûã òîäîðõîéëîõûí òóëä y(3) = 7 ãýñýí àíõíû í°õöëèéã îðëóóëàõàä. 33 − 3 + C = 7, ⇒ 6 + C = 7, ⇒ C = 1. 3 Òèéìýýñ òóõàéí í°õö°ëèéã õàíãàõ øèéä y = x3 − x + 1. 3 Æèøýý 6. dy = 2x − 1 x = 0, y (1) = 5 òóõàéí øèéäèéã îë. dx x2 , x2 = 2x 1 = − 1 x x2 Äèôôåðåíöèàë òýãøèòãýëèéí åð°íõèé øèéä y = x2 + 1 + C. x Æèøýý 7. dy 2 = , y (0) = 2 òóõàéí øèéäèéã îë. dx x + 1 2 ýõ ôóíêö íü y = 2 ln |x + 1| + C x+1 C-èéí óòãûã òîäîðõîéëîõûí òóëä y(0) = 2 ãýñýí àíõíû í°õöëèéã îðëóóëàõàä. 2 ln |0 + 1| + C = 2, ⇒ 2 · ln 1 + C = 2, ⇒ 2 · 0 + C = 2, ⇒C =2

0.1. Èíòåãðàëûí áîäëîãî 6 Òèéìýýñ òóõàéí í°õö°ëèéã õàíãàõ øèéä y = 2 ln |x + 1| + 2. Æèøýý 8. dr θ π = cos , r = 2 òóõàéí øèéäèéã îë. dθ 2 3 cos θ ýõ ôóíêö íü r (θ) = 2 sin θ + C. 2 2 π = 2 ãýñýí àíõíû C-èéí óòãûã òîäîðõîéëîõûí òóëä r 3 í°õöëèéã îðëóóëàõàä π ⇒ 2 sin π + C = 2, ⇒ 2 · 1 + C = 2, 2 sin 3 + C = 2, 2 62 ⇒ 1 + C = 2, ⇒ C = 1 Òèéìýýñ òóõàéí í°õö°ëèéã õàíãàõ øèéä θ r (θ) = 2 sin + 1. 2 Æèøýý 9. dz = cos t − 2 sin t, z (0) = 5 òóõàéí øèéäèéã îë. dt cos t − 2 sin t ýõ ôóíêö íü z (t) = sin t + 2 cos t + C. C-èéí óòãûã òîäîðõîéëîõûí òóëä z(0) = 5 ãýñýí àíõíû í°õöëèéã îðëóóëàõàä sin 0 + 2 cos 0 + C = 5, ⇒ 0 + 2 · 1 + C = 5, ⇒ C = 3. Òèéìýýñ òóõàéí í°õö°ëèéã õàíãàõ øèéä z (t) = sin t + 2 cos t + 3.

Ãàð÷èã 7 Æèøýý 10. dy 1 y(x) ôóíêöèéã = + 2x äèôôåðåíöèàë òýãøèòãýëýýð dx x °ã°ãäñ°í. y(1) = 0, x = e ôóíêöèéí óòãûã îë. 1 + 2x ýõ ôóíêö íü y = ln x + x2 + C. x C-èéí óòãûã òîäîðõîéëîõûí òóëä y(1) = 0 ãýñýí àíõíû í°õöëèéã îðëóóëàõàä y (1) = 0, ⇒ ln 1 + 12 + C = 0, ⇒ 0 + 1 + C = 0, ⇒ C = −1. Òèéìýýñ òóõàéí í°õö°ëèéã õàíãàõ øèéä y = ln x + x2 − 1. x = e óòãûã òîîöîîëîõîä y (e) = ln e + e2 − 1 = 1 + e2 − 1 = e2.  Æèøýý 11. íä°ð áàðèëãûí äýýðýýñ óíàñàí á°ìá°ã v(t) = −10t−5m/c õóðäòàé óíàñàí. t = 4ñåê ãàçàð óíàñàí áîë áàðèëãûí °íä- ðèéã îë. H áàðèëãûí °íä°ð.Á°ìá°ãíèé îéëòûã h(t) ôóíêöýýð. Õóðä- íû òýãøèòãýëèéí ýõ ôóíêö íü áàðèëãûí °íäðèéã îëîõ òýã- øèòãýë áîëíî. v(t) = −10t − 5m/c ýõ ôóíêö íü h(t) = C − 5t2 − 5t C îëîõûí òóëä h(0) = H í°õöëèéã àøèãëàíà. h (t) = H − 5t2 − 5t.

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 8 t = 4ñåê äàðàà °íä°ð íü 0 òýíöýíý. H − 5 · 42 − 5 · 4 = 0, ⇒ H = 80 + 20 = 100 m. 0.2 Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 0.2.1 Òîäîðõîéã³é èíòåãðàë f (x) ôóíêö õÿçãààðã³é ýõ ôóíöòýé á°ã°°ä á³ãä ç°âõ°í òîãòìîë C ÿëãààòàé áàéäàã. (F (x) + C) = F (x) + C = f (x) + 0 = f (x) . f (x) ôóíêöèéí ýõ ôóíêöèéã òîäîðõîéã³é èíòåãðàë ãýæ íýðëýäýã á°ã°°ä äàðààõ áàéäëààð òýìäýãëýíý. f (x)dx = F (x) + C, F (x) = f (x) . Ýíý òîäîðõîéëîëòîä íü èíòåãðàëûí òýìäýã, f (x) íü ôóíêö, x íü èíòåãðàëûí õóâüñàã÷, dx íü x õóâü- ñàã÷èéí äèôôåðåíöèàë, C íü èíòåãðàëûí òîãòìîë òîî.

Ãàð÷èã 9 0.2.2 Çàðèì ôóíêöèéí èíòåãðàë Ýíä a, p(= 1), C áîäèò òîãòìîë òîî, b çýðýãò ôóíêöèéí ñóóðü òóë (b = 1, b > 0) adx = ax + C x2 x2dx = x3 + C xpdx = xp+1 + C dx = ln |x| + C xdx = + C 3 p+1 x 2 sin xdx = cos xdx = sin x + C exdx = ex + C − cos x + C tg xdx = ctg xdx = sec xdx = sec2xdx = tg x + C −ln |cos x| + C ln |sin x| + C ln tg x + π +C = 2 4 ln |sec x + tg x| + C csc xdx = csc2xdx = sec x tg xdx = csc x ctg xdx = − ctg x + C sec x + C − csc x + C ln tg x +C = 2 − ln |csc x + ctg x| + C dx = dx = dx = dx = 1+x2 = a2 +x2 1−x2 a2 −x2 arctg x + C 1 arctg x + C 1 ln 1+x +C 1 ln a+x +C a a 2 1−x 2a a−x √ dx 1−x2 √ dx = √ dx = √dx = a2 −x2 x x2−1 arcsin x + C x √ x2±a2 arcsin a + C ln x + x2 ± a2 + arcsec |x| + C C sinh xdx = cosh xdx = sech2xdx = csch2xdx = cosh x + C sinh x + C −ctgh x + C tg hx + C sech x tg hxdx = csch x ctg hxdx = tg hxdx = −sech x + C −csch x + C ln cosh x + C 0.2.3 Òîäîðõîéã³é èíòåãðàëûí ÷àíàðóóä 1 Õýðýâ a òîãòìîë òîî áîë af (x) dx = a f (x) dx, Òîãòìîë òîî(êîýôôèöèåíòèéã) èíòåãðàë òýìäãèéí °ìí° àâ÷ áîëíî.

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 10 2 f (x), áà g(x) ôóíêöèéí õóâüä g (x) dx, [f (x) ± g (x)] dx = f (x) dx ± Íèéëáýð (ÿëãàâàð)-ûí òîäîðõîéã³é èíòåãðàë íü èíòåãðà- ëûí íèéëáýð (ÿëãàâàð)-òýé òýíö³³ áàéíà. Æèøýý 12. 3x2 − 6x + 2 cos x dx èíòåãðàëûã áîä. 1 áà 2-ð ÷àíàðûã àøèãëàõàä 6xdx I = 3x2 − 6x + 2 cos x dx = 3x2dx − + 2 cos xdx = 3 x2dx − 6 xdx + 2 cos xdx. Ãóðâàí èíòåãðàëûã õ³ñíýãòèéã àøèãëàí áîäîõîä. I = 3 · x3 − 6 · x2 + 2 · sin x + C = x3 − 3x2 + 2 sin x + C. 3 2 Æèøýý 13. (1 + x) (1 + 2x) dx èíòåãðàëûã áîä. (1 + x) (1 + 2x) = 1 + x + 2x + 2x2 = 2x2 + 3x + 1. (1 + x) (1 + 2x) dx = 2x2 + 3x + 1 dx = 2x2dx+ 3xdx+ 1dx = 2 x2dx+3 xdx+ dx

Ãàð÷èã 11 x3 x2 2x3 3x2 = 2 · + 3 · + x + C = + + x + C. 32 32 Æèøýý 14. 11 − dx èíòåãðàëûã áîä. x2 x3 I= 1 − 1 dx = dx − dx x2 x3 x2 x3 . I= x−2dx − x−3dx = x−1 − x−2 + C = −1 + 1 + C (−1) (−2) x 2x2 Æèøýý 15. √√ x + 3 x dx òîîöîîë. √√ √√ x + 3 x dx = xdx + 3 xdx 1 +1 1 +1 x x1 1 2 3 = x2 dx + x3 dx = 1 +1 + 1 +1 +C √ 2 √ 3 2x 3 3x 4 2 x3 3 3 x4 2 3 = + = + + C. 34 3 4 Æèøýý 16. x√+ 1 èíòåãðàëûã áîä. x dx x√+ 1 = √x + √1 dx = √ + √1 dx x dx xx x x

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 12 = √ √dx = x3 √ = √ √ xdx + x 2 +2 x+C 2 x3 + 2 x + C. 3 3 2 Æèøýý 17. √ 2dx èíòåãðàëûã áîä. x+ x I= √ 2dx = x2 + √ + √ 2 dx x+ x 2x x x = x2 + 3 + x dx. 2x 2 I= x2 + 3 + x dx = x2dx + 2 3 xdx. 2x 2 x2 dx + x3 x5 x2 x3 4x 5 x2 I = +2· 2 2 3 √ 5 + +C = + 5 + +C 2 23 2 x3 4 x5 x2 =+ + + C. 352 Æèøýý 18. 3 + √2 dx èíòåãðàëûã áîä. √ 3x x √3 + √2 dx = 3√dx + 2√dx 3x x 3x x x− 1 x− 1 x− 1 +1 x− 1 +1 3 2 3 2 · · =3 dx + 2 dx = 3 − 1 + 1 + 2 − 1 + 1 + C 3 2 √ 9x 2 9 3 x2 √ 3 1 = + 4x2 + C = + 4 x + C. 22

Ãàð÷èã 13 Æèøýý 19. √ + e3 dx èíòåãðàëûã áîä. 3x I= √ + e3 dx = 1 + e3 dx 3x x3 = 1 e3dx = 1 dx + e3 dx. x3 dx + x3 x4 √ 3 3 3 x4 I= 1 dx + e3 dx = + e3x + C = 4 + e3x + C. 4 x3 3 Æèøýý 20. 4dx 2 + 3x2 òîîöîîë. dx 1 x a2 + x2 = a arctg a + C. 4dx dx 4 dx 2 + 3x2 = 4 = 3 2 + x2 3 2 2 3 3 + x2 √ = 4 · 1 arctg x + C = √4 arctg √3x + C. 32 62 2 33 Æèøýý 21. x2 1 + x2 dx èíòåãðàëûã áîä. Èíòåãðàëûã íèéëáýðò çàäëàõàä

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 14 x2 1 + x2 − 1 1 + x2 − 1 dx I = 1 + x2 dx = 1 + x2 dx = 1 + x2 1 + x2 = 1 − 1 1 x2 dx = dx − dx + 1 + x2 . I= dx − dx = x − arctg x + C. 1 + x2 Æèøýý 22. dx 1 + 2x2 èíòåãðàëûã áîä. I= dx dx 1 dx 1 + 2x2 = = 2 1 + x2 2 1 + x2 2 2 1 dx = 2. 2 √1 + x2 2 dx 1 x = arctg , a2 + x2 a a 1 dx =1· 1x +C I= 2 √1 arctg √1 √1 2 2 22 + x2 2 √ 2 √ + C = √1 arctg √ + C. 2x 2x = arctg 22 Æèøýý 23. √ πdx èíòåãðàëûã áîä. π − x2

Ãàð÷èã 15 √ dx x = arcsin + C, a2 − x2 a √ πdx = π dx = π arcsin √x + C. π − x2 (√π)2 π − x2 Æèøýý 24. (2 cos x − 5 sin x) dx èíòåãðàëûã áîä. (2 cos x − 5 sin x) dx = 2 cos xdx − 5 sin xdx = 2 cos xdx − 5 sin xdx = 2 · sin x − 5 · (− cos x) + C = 2 sin x + 5 cos x + C. Æèøýý 25. dx èíòåãðàëûã áîä. 1 − x2 2 Àëãåáðèéí õÿëáàð÷ëàõ ³éëäýë õèéõýä I= dx dx dx √ = = √ =2 1 − x2 1 (2 − x2) 1 2 − x2 2 2 2 √ dx √ dx . =2 2 − x2 2 − x2 √ 2 √ dx x = arcsin , a2 − x2 a

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 16 √ dx = √ √x + C. I= 2 2 arcsin 2 √ 2 2 − x2 Æèøýý 26. tg2xdx èíòåãðàëûã áîä. tg2x = sec2x − 1, tg2xdx = sec2x − 1 dx = sec2xdx − dx = tg x − x + C Æèøýý 27. ctg2xdx èíòåãðàëûã áîä. 1 − ctg2x = 1, ⇒ ctg2x = 1 − 1. sin2x sin2x I = ctg2xdx = 1 dx dx. sin2x − 1 dx = sin2x − I= dx − dx = − ctg x − x + C. sin2x Æèøýý 28. dx sin22x îðëóóëãûã àøèãëàõã³éãýýð èíòåãðàëûã áîä. sin 2x = 2 sin x cos x, sin2x + cos2x = 1 dx 1 dx 1 sin2x + cos2x dx sin22x = 4 sin2xcos2x = 4 sin2xcos2x

Ãàð÷èã 17 1 11 1 sec2xdx + 1 csc2xdx = cos2x + sin2x dx = 4 4 4 = 1 tg x − 1 ctg x + C = 1 (tg x − ctg x) + C. 44 4

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 18 0.2.4 Îðëóóëàõ àðãààð áîäîõ Ýíý õýñýãò áèä îëîí ò°â°ãòýé èíòåãðàëóóäûã áîäîõ ÷óõàë àðãûã ³çýõ áîëíî. f (u) ýõ ôóíêö íü F (u) áàéíà ãýæ ³çüå. f (u) du = F (u) + C. u = u(x), d F (u (x)) = F (u (x)) u (x) = f (u (x)) u (x) . dx Õî¼ð òàëààñ íü èíòåãðàë àâàõàä f (u (x)) u (x) dx = F (u (x)) + C. f (u (x))u (x)dx = f (u) du, u = u (x). Òîäîðõîéã³é èíòåãðàëûí îðëóóëãûí òîìü¼î. Æèøýý 29. x e 2 dx èíòåãðàëûã òîîöîîë. x u = 2 îðëóóëàõàä. dx ⇒ dx = 2du. du = , 2 x eu · 2du = 2 eudu = 2eu + C = x + C. e 2 dx = 2e 2 Æèøýý 30. (3x + 2)5dx èíòåãðàëûã áîä.

Ãàð÷èã 19 u = 3x + 2 îðëóóëãûã õèéõýä du = d (3x + 2) = 3dx. du dx = 3 (3x + 2)5dx = u5 du = 1 u5du 33 1 u6 u6 (3x + 2)6 = · +C = +C = + C. 36 18 18 Æèøýý 31. √ dx èíòåãðàëûã áîä. 1 + 4x u = 1 + 4x, du = d(1 + 4x) = 4x. du du = d (1 + 4x) = 4dx, dx = 4 √ dx = du 1 √du 1 u− 1 du = = 2 √4 1 + 4x u4 u4 √ = 1· u1 +C = 1 · 2u 1 +C = u1 +C = u +C 4 2 4 2 2 2 √ 2 1 2 1 + 4x = + C. 2 Æèøýý 32. √ xdx èíòåãðàëûã áîä. 1 + x2 u = 1 + x2, du = d 1 + x2 = 2xdx. du xdx = 2

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 20 √xdx = du d√u √ 1 + x2 + C. = u+C = √2 = 1 + x2 u 2u Æèøýý 33. √ dx èíòåãðàëûã òîîöîîë. a2 − x2 x u = , x = au, dx = adu. a √ dx = adu adu a2 − x2 = a2 (1 − u2) a2 − (au)2 = √&adu = √ du x = arcsin u + C = arcsin + C. &a 1 − u2 1 − u2 a Æèøýý 34. x2 dx îðëóóëãûã àøèãëàí áîä. 1 x3 + u = x3 + 1, du = d x3 + 1 = 3x2dx. x2dx = du I= x2 du du , x3 + 1dx = . 3 3= u 3u I= du 1 du = 1 ln |u| + C. = 3u 3 u 3 I = 1 ln |u| + C = 1 ln x3 + 1 + C. 33 Æèøýý 35. √ 3 1 − 3xdx èíòåãðàëûã áîä.

Ãàð÷èã 21 u = 1 − 3x, du = d (1 − 3x) = −3dx, dx = −du. 3 √ √ du 1 √ 3 1 − 3xdx = 3u − =− 3 udu 33 = −1 1 −1 · u4 −1 · 3u 4 3 3 3 3 3 u3 du = 4 +C = 4 +C 3 u4 √ 3 (1 − 3x)4 3 3 u4 =− +C =− +C =− + C. 44 4 Æèøýý 36. x+1 dx èíòåãðàëûã áîä. 5 x2 + 2x − u = x2 + 2x − 5, du = 2xdx + 2dx = 2(x + 1)dx, du (x + 1)dx = . 2 Èíòåãðàëûã øèíý õóâüñàã÷ààð áîäîõîä õÿëáàð áîëíî. x+1 dx = du = 1 du = 1 ln |u| + C 5 2 u2 x2 + 2x − u2 = 1 x2 + 2x − 5 + C. ln 2 Æèøýý 37. xdx 1 + x4 èíòåãðàëûã áîä. u = x2, du = 2xdx, ⇒ xdx = du . 2

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 22 xdx du = 1 du 1 1 + x4 = 2 2 1 + u2 = 2 arctg u + C 1 + u2 = 1 arctg x2 + C. 2 Æèøýý 38. xdx x4 + 2x2 + 1 èíòåãðàëûã áîä. x4 + 2x2 + 1 = x2 + 1 2 = 1 + x2 2, u = 1 + x2, du = d 1 + x2 du = 2xdx, xdx = . 2 xdx xdx du = 1 du x4 + 2x2 + 1 = (1 + x2)2 = 2 2 u2 u2 1 u−2du = 1 · u−1 + C = −1 + C = − (1 1 x2) + C. = 2 (−1) 2u 2 + 2 Æèøýý 39. 2xexdx èíòåãðàëûã áîä. 2xexdx = (2e)xdx. 2e = a, axdx = ax + C = (2e)x + C (2e)xdx = ln a ln (2e) = 2xex +C = 2xex + C. ln 2 + ln e ln 2 + 1 Æèøýý 40.

Ãàð÷èã 23 xe−x2dx èíòåãðàëûã áîä. u = −x2, du = d −x2 = −2xdx, xdx = −du 2 xe−x2dx = eu −du = −1 22 eudu = 1eu + C 2 e−x2 = − + C. 2 Æèøýý 41. sin x dx èíòåãðàëûã áîä. 1 − cos x u = 1 − cos x, du = − (− sin x) dx = sin xdx. sin x du = ln |u| + C = ln |1 − cos x| + C. 1 − cos xdx = u Æèøýý 42. ⇒ du = 2udu. u2 − 1 u2du √ x x + 1dx èíòåãðàëûã áîä. √ u = x + 1, u2 = x + 1, ⇒ x = u2 − 1, √ x x + 1dx = u2 − 1 u · 2udu = 2 = 2 u4 − u2 du = 2 u4du − 2 u2du u5 u3 2 52 3 = 2 · − 2 · + C = (x + 1)2 − (x + 1)2 + C. 53 5 3 Æèøýý 43.

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 24 ctg (3x + 5) dx èíòåãðàëûã áîä. ctg (3x + 5) dx = cos (3x + 5) dx. sin (3x + 5) u = sin (3x + 5) , du = 3 cos (3x + 5) dx, ⇒ cos (3x + 5) dx = du 3 ctg (3x + 5) dx = cos (3x + 5) du dx = 3 sin (3x + 5) u = 1 du = 1 ln |u| + C = 1 ln |sin (3x + 5)| + C. 3u3 3 Æèøýý 44. √ sin 2x dx èíòåãðàëûã áîä. 1 + cos2x u = 1 + cos2x, ⇒ du = 1 + cos2x dx = 2 cos x · (− sin x) dx = − sin 2xdx. √ sin 2x dx = (−√du) = −2 d√u 1 + cos2x u 2u √ 1 + cos2x + C. = −2 u + C = −2

Ãàð÷èã 25 0.2.5 Õýñýã÷ëýí èíòåãðàë÷ëàõ u(x), v(x) ÿëãààòàé ôóíêö áàéíà ãýæ ³çüå. d (uv) = udv + vdu. udv = d (uv) − vdu. udv = uv − vdu. Æèøýý 45. x sin (3x − 2) dx èíòåãðàëûã áîä. udv = uv − vdu, u = x, dv = sin (3x − 2) dx. v= sin (3x − 2) dx = − 1 cos (3x − 2) , du = dx. 3 x sin (3x − 2) dx = = −x cos (3x − 2) − −1 cos (3x − 2) dx 3 3 = −x cos (3x − 2) + 1 cos (3x − 2) dx 33 = −x cos (3x − 2) + 1 · 1 sin (3x − 2) + C 3 33 = 1 sin (3x − 2) − x cos (3x − 2) + C. 93 Æèøýý 46.

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 26 xcsc2xdx èíòåãðàëûã áîä. u = x, du = dx dv = csc2 xdx = dx sin2 x dx v= dv = sin2x = − ctg x. udv = uv − vdu, ⇒ x csc2xdx = x (− ctg x) − (− ctg x) dx u dv uv v du = −x ctg x + ctg xdx. ctg xdx = ln |sin x| + C. xcsc2xdx = −x ctg x + ln |sin x| + C. Æèøýý 47. x cos 2xdx èíòåãðàëûã áîä. u = x, dv = cos 2xdx du = dv, 1 v = cos 2xdx = sin 2x. 2 x cos 2xdx = x · 1 sin 2x − 1 sin 2xdx 22

Ãàð÷èã 27 = x sin 2x − 1 sin 2xdx = x sin 2x − 1 · −1 cos 2x +C 22 22 2 x1 = sin 2x + cos 2x + C. 24 Æèøýý 48. ln xdx òîîöîîë. u = ln x, dv = dx 1 du = dx, v = dx = x. x ln xdx = x ln x − x · 1 dx = x ln x − x + C. x Æèøýý 49. ln x x2 dx èíòåãðàëûã áîä. dx u = ln x, dv = x2 . dx dx = −1. du = , v = x2 x x ln x = ln x · −1 − − 1 dx x2 dx x xx = −ln x + dx = −ln x − 1 + C. x x2 x x Æèøýý 50. log2xdx èíòåãðàëûã áîä.

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 28 log2xdx = 1 · log2xdx, u = log2x, dv = 1dx du = dx , v = 1dx = x x ln 2 log2xdx = xlog2x − x · dx x ln 2 1x = xlog2x − ln 2 dx = xlog2x − ln 2 + C. Æèøýý 51. x2xdx èíòåãðàëûã áîä. u = x, dv = 2xdx, du = dx, v = 2xdx = 2x ln 2 x2xdx = x2x − 2x dx = x2x − 1 2xdx ln 2 ln 2 ln 2 ln 2 x2x 1 2x x2x 2x = ln 2 − ln 2 · ln 2 + C = ln 2 − (ln 2)2 + C 2x 1 = x − + C. ln 2 ln 2 Æèøýý 52. xe−xdx èíòåãðàëûã áîä. u = x, dv = e−xdx, du = dx, v = e−xdx = −e−x.

Ãàð÷èã 29 xe−xdx = x −e−x − −e−x dx = −xe−x + e−xdx = −xe−x − e−x + C = −e−x (x + 1) + C. Æèøýý 53. x2exdx èíòåãðàëûã áîä. u = x2, dv = exdx. du = 2xdx, v = exdx = ex, x2exdx = x2ex − 2xexdx = x2ex − 2 xexdx. ѳ³ëèéí èíòåãðàëûã äàõèí õýñýã÷èëáýë u = x, dv = exdx, du = dx, v = exdx = ex, x2exdx = x2ex − 2 xexdx = x2ex − 2 xex − exdx = x2ex − 2 (xex − ex) + C = x2ex − 2xex + 2ex + C = ex x2 − 2x + 2 + C. Æèøýý 54. arcsin xdx èíòåãðàëûã áîä. u = arcsin x, dv = dx

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 30 du = d arcsin x = √ dx , v = dx = x dx 1 − x2 arcsin xdx = x arcsin x − √ x dx. 1 − x2 w = 1 − x2, dw = −2xdx, xdx = −dw 2 √ x dx = −√d2w = − d√w √ 1 − x2 w 2w =− w+C = − 1 − x2 + C. arcsin xdx = x arcsin x − − 1 − x2 + C = x arcsin x + 1 − x2 + C. Æèøýý 55. arctg xdx èíòåãðàëûã áîä. u = arctg x, dv = dx. dx du = 1 + x2 , v = x, I= arctg xdx = x arctg x − xdx 1 + x2 . t = x2 + 1, ⇒ dt = 2xdx, ⇒ xdx = dt , 2 xdx dt = 1 dt = 1 ln |t| 1 + x2 = 2 t2 t2

Ãàð÷èã 31 1 1 + x2 1 1 + x2 . = ln = ln 22 I = x arctg x − 1 ln 1 + x2 + C. 2 Æèøýý 56. ex sin xdx èíòåãðàëûã áîä. udv = uv − vdu, u = ex, dv = sin xdx. du = exdx, v = sin xdx = − cos x, ex sin xdx = −ex cos x + ex cos xdx. u = ex, dv = cos xdx, du = exdx, v = cos xdx = sin x ex sin xdx = −ex cos x + ex cos xdx = −ex cos x + ex sin x − ex sin xdx. 2 ex sin xdx = ex sin x − ex cos x ex sin xdx = ex (sin x − cos x) + C. 2 Æèøýý 57. e−x sin xdx èíòåãðàëûã áîä.

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 32 u = sin x, dv = e−xdx du = cos xdx, v = e−xdx = −e−x, e−x sin xdx = −e−x sin x − −e−x cos xdx = −e−x sin x + e−x cos xdx. u = cos x, dv = e−xdx. du = − sin xdx, v = e−xdx = −e−x. e−x sin xdx = −e−x sin x + e−x cos xdx = −e−x sin x + −e−x cos x − −e−x (− sin x) dx = −e−x sin x − e−x cos x − e−x sin xdx. e−x sin xdx = −e−x (sin x + cos x) . 2 Æèøýý 58. sin2xdx èíòåãðàëûã áîä. u = sin2x, dv = dx. du = 2 sin x cos xdx = sin 2xdx, v = dx = x I = sin2xdx = x sin2x − x sin 2xdx.

Ãàð÷èã 33 u = x, dv = sin 2xdx, du = dx, v = sin 2xdx = −1 cos 2x. 2 x sin 2xdx = −x cos 2x − −1 cos 2x dx 2 2 = −x cos 2x + 1 cos 2xdx = −x cos 2x + 1 · 1 sin 2x + C 22 2 22 = −x cos 2x + 1 sin 2x + C. 24 I = x sin2x − x sin 2xdx = x sin2x + x cos 2x − 1 sin 2x + C. 24 cos 2x = cos2x − sin2x, sin2x + cos2x = 1. I = x sin2x + x cos2x − sin2x − 1 sin 2x + C 24 = x sin2x + xcos2x − xsin2x − 1 sin 2x + C 2 24 x sin2x + cos2x − 1 sin 2x + C = x − 1 sin 2x + C. = 2 4 24 Æèøýý 59. cos2xdx èíòåãðàëûã áîä. u = cos2x, dv = dx. du = −2 cos x sin xdx = − sin 2xdx, v = dx = x.

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 34 I = cos2xdx = x cos2x − x (− sin 2x) dx = x cos2x + x sin 2xdx. u = x, dv = sin 2xdx. du = dx, v = sin 2xdx = −1 cos 2x, 2 x sin 2xdx = −x cos 2x − −1 cos 2x dx 2 2 = −x cos 2x + 1 cos 2xdx = −x cos 2x + 1 · 1 sin 2x + C 22 2 22 = −x cos 2x + 1 sin 2x + C. 24 I = x cos2x + x sin 2xdx = x cos2x − x cos 2x + 1 sin 2x + C. 24 cos 2x = cos2x − sin2x, sin2x + cos2x = 1. I = x cos2x − x cos2x − sin2x 1 + sin 2x + C 24 = x cos2x − xcos2x + xsin2x + 1 sin 2x + C 2 24 x cos2x + sin2x 1 x1 = + sin 2x + C = + sin 2x + C. 2 4 24 Æèøýý 60.

Ãàð÷èã 35 sinnxdx, n ≥ 2 òîìü¼îã îëîîðîé. udv = uv − vdu, u = sinn−1 x, dv = sin xdx. du = d sinn−1x = (n − 1) sinn−2x cos xdx, dx v = sin xdx = − cos x. sinnxdx = = − cos x sinn−1x − (− cos x) (n − 1)sinn−2x cos xdx = − cos x sinn−1x − (n − 1) sinn−2xcos2xdx = − cos x sinn−1x + (n − 1) sinn−2x 1 − sin2x dx = − cos x sinn−1x+(n − 1) sinn−2xdx−(n − 1) sinnxdx. sinnxdx + (n − 1) sinnxdx = = − cos x sinn−1x + (n − 1) sinn−2xdx, ⇒ n sinnxdx = (n − 1) sinn−2xdx − cos x sinn−1x, ⇒ sinnxdx = n − 1 sinn−2xdx − cos x sinn−1x nn

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 36 0.2.6 ax2 + bx + c õýëáýðèéí èíòåãðàë áîäîõ ax2 + bx + c = a(u2 + k), b 4ac − b2 u=x+ , k= 2a 4a2 1. √ dx = ln x + x2 ± a2 x2 ± a2 2. √ dx x = arcsin a2 − x2 a dx 1 x 3. a2 + x2 = a arctg a dx 1 a + x 4. = ln a2 − x2 2a a−x Æèøýý 61. dx x2 − 5x + 7 èíòåãðàëûã áîä. x2 − 5x + 7 = x2 − 2 · 5 + 5 2 52 x +7 22 − 2 5 2 25 5 2 28 − 25 = x− +7− = x− + 2 4 24 = x− 5 2 √2 3 + . 22 u = x− 5 du = dx , 2

Ãàð÷èã 37 dx dx √ 2= du √ 2. x2 − 5x + 7 = x − 5 2+ 3 u2 + 3 2 2 2 du = 1 arctg u +C = √2 arctg x − 5 +C 2 √ √ √ √ 2 3 3 33 u2 + 3 2 22 2 = √2 arctg 2x√− 5 + C. 33 Æèøýý 62. dx x2 − x + 2 èíòåãðàëûã áîä. x2 − x + 2 = x2 − 2 · x · 1 + 12 1 2 − 22 2 +2 = x−1 2−1+2= x− 1 27 24 + 24 √2 = x− 1 2 7 . + 22 u = x − 1 , du = dx, 2 dx dx √ 2= du x2 − x + 2 = 2+ 7 √2 − 1 2 u2 + 7 x 2 2 1 u + C = √2 arctg x− 1 + C =√ arctg √ 2 √ 77 77 22 2 = √2 arctg 2x√− 1 + C. 77

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 38 Æèøýý 63. dx x2 + 10x + 26 èíòåãðàëûã áîä. x2 + 10x + 26 = x2 + 10x + 25 + 1 = (x + 5)2 + 1, dx dx I = x2 + 10x + 26 = (x + 5)2 + 1. u = x + 5, du = dx, du I = u2 + 1 = arctg u + C = arctg (x + 5) + C. Æèøýý 64. dx 5 − 4x − x2 èíòåãðàëûã áîä. 5 − 4x − x2 = 5 − x2 + 4x = 9 − x2 + 4x + 4 = 9 − (x + 2)2 = 32 − (x + 2)2. dx dx I = 5 − 4x − x2 = 32 − (x + 2)2 . u = x + 2, du = dx, I= du 1 3 + u 1 3+x+2 32 − u2 = 2 · 3 ln + C = ln +C 3−u 6 3 − (x + 2) 1 5+x + C. = ln 6 1−x Æèøýý 65. √ dx èíòåãðàëûã áîä. x2 + x − 2

Ãàð÷èã 39 x2 + x − 2 = x2 + x + 1 − 1 −2 = 12 9 44 x+ − 24 = 1 2 32 x+ . − 22 1 u = x + , du = dx, 2 √ dx = dx du = x2 + x − 2 2− 32 32 x + 1 2 u2 − 2 2 = ln u + u2 − 32 1 x2 + x − 2 + C. + C = ln x + + 22 Æèøýý 66. dx 8 − 2x − x2 èíòåãðàëûã áîä. 8 − 2x − x2 = 8 − x2 + 2x = 9 − x2 + 2x + 1 = 32 − (x + 1)2. dx dx I = 8 − 2x − x2 = 32 − (x + 1)2 . u = x + 1, du = dx, dx du 1 3 + u I = 32 − (x + 1)2 = 32 − u2 = 2 · 3 ln 3 − u + C 1 3+x+1 1 4+x = ln + C = ln + C. 6 3 − (x + 1) 6 2−x

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 40 Æèøýý 67. √ dx èíòåãðàëûã áîä. 1 − 2x − x2 1 − 2x − x2 = 1 − x2 + 2x = 2 − x2 + 2x + 1 = 2 − (x + 1)2 = √ 2 − (x + 1)2. 2 u = x + 1, du = dx √ dx = dx = du 1 − 2x − x2 √ 2 − (x + 1)2 √ 2 − u2 2 2 = arcsin √u + C = arcsin x√+ 1 + C. 22 Æèøýý 68. x+1 dx èíòåãðàëûã áîä. 1 x2 + x + x2 + x + 1 = x2 + x + 1 + 3 = 1 2 √2 44 x+ 3 2 + . 2 11 x+1= x+ + . 22 x+1 dx = x + 1 + 1 dx 1 x2 + 2 2 x2 + x + x +1 = x2 x+ 1 dx + 1 dx +x 2 1 2 x2 + x + 1 +

Ãàð÷èã 41 1 (2x + 1) dx 1 dx = x2 + x + 1 + 2 2 x + 1 2+ √2 2 3 2 1 d x2 + x + 1 1 d x + 1 = + 2 2 2 √2 x2 + x + 1 x + 1 2+ 3 2 2 = 1 x2 + x + 1 +1· 1 arctg x+ 1 +C ln 2 √ √ 2 23 3 22 1 x2 + x + 1 + √1 arctg 2x√+ 1 + C. = ln 2 33 Æèøýý 69. xdx x2 + 2x + 10 èíòåãðàëûã áîä. x2 + 2x + 10 = x2 + 2x + 1 + 9 = (x + 1)2 + 32. x=x+1−1 x+1−1 xdx dx x2 + 2x + 10 I = x2 + 2x + 10 = = (x + 1) dx − dx x2 + 2x + 10 x2 + 2x + 10 1 (2x + 2) dx − (x + dx + 32 = I1 − I2. = x2 + 2x + 10 1)2 2 u = x2 + 2x + 10, du = (2x + 2)dx 1 (2x + 2) dx 1 du = 1 ln |u| I1 = 2 x2 + 2x + 10 = 2 u2

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 42 1 x2 + 2x + 10 1 x2 + 2x + 10 . = ln = ln 22 dx 1 x + 1 I2 = (x + 1)2 + 32 = 3 arctg 3 . 1 x2 + 2x + 10 − 1 arctg x + 1 + C. I = I1 − I2 = 2 ln 33 Æèøýý 70. èíòåãðàëûã áîä. √ xdx 5 + x − x2 5 + x − x2 = 1 − 2x, I = √ xdx = −1 √ (−2x) dx 5 + x − x2 2 5 + x − x2 = −1 (√−2x + 1 − 1) dx 2 5 + x − x2 = −1 √(1 − 2x) 1 √ dx = I1 + I2. 2 dx + 5 + x − x2 5 + x − x2 2 u = 5 + x − x2, du = (1 − 2x) dx. 1 √(1 − 2x) dx = −1 √du √ I1 = − =− u 2 5 + x − x2 2 u = − 5 + x − x2. 5 + x − x2 = 5 − x2 − x = 5 + 1 − x2 − x + 1 44

Ãàð÷èã 43 = 21 − x− 1 2 √2 x− 1 2 21 − 2 = . 4 22 1 √ dx 1 dx I2 = 2 = 5 + x − x2 2 √ 2 21 x − 1 2 2 − 2 = 1 x− 1 = 1 arcsin 2√x − 1. arcsin √ 2 2 21 2 21 2 I = − 5 + x − x2 + 1 arcsin 2√x − 1 + C. 2 21 Æèøýý 71. √ x+1 dx èíòåãðàëûã áîä. x2 + 4x + 8 x2 + 4x + 8 = 2x + 4. √ x+1 1 √ 2x + 2 dx dx = x2 + 4x + 8 2 x2 + 4x + 8 1 √2x + 4 − 2 1 √ 2x + 4 dx = dx = 2 x2 + 4x + 8 2 x2 + 4x + 8 − √ dx + 8 = I1 − I2. x2 + 4x u = x2 + 4x + 8, du = (2x + 4) dx. 1 √ 2x + 4 1 √du √ x2 + 4x + 8. I1 = 2 dx = = u= x2 + 4x + 8 2 u x2 + 4x + 8 = x2 + 4x + 4 + 4 = (x + 2)2 + 22.

0.2. Òîäîðõîéã³é èíòåãðàë ³íäñýí ä³ðì³³ä 44 t = x + 2, dt = dx, I2 = √ dx = dx √ dt x2 + 4x + 8 = t2 + 22 (x + 2)2 + 22 = ln t + t2 + 22 = ln x + 2 + x2 + 4x + 8 . I = x2 + 4x + 8 − ln x + 2 + x2 + 4x + 8 + C.

Ãàð÷èã 45 0.3 Ðàöèîíàë ôóíêöèéí èíòåãðàë 0.3.1 Òîäîðõîé áóñ êîýôôèöèåíòèéí àðãààð çàäëàõ f (x) = P (x) , Q(x) = 0 ðàöèîíàë ôóíêöèéí õýëáýð Q (x) Ðàöèîíàë ôóíêöèéí íèéëáýðèéã àâ÷ ³çüå. 2 3 2(x + 4) + 3(x + 1) 2x + 8 + 3x + 3 += = x2 + x + 4x + 4 x+1 x+4 (x + 1)(x + 4) = 5x + 11 . 4 x2 + 5x + Òîäîðõîé áóñ êîýôôèöèåíòèéí àðãà àøèãëàí ðàöèîíàë áó- òàðõàéã õÿëáàð ðàöèîíàë áóòàðõàéí íèéëáýðò çàäàë. 71-81 Æèøýý 72. 6 (x + 2) (x − 4). 6 AB = + . (x + 2) (x − 4) x + 2 x − 4 A B A (x − 4) + B (x + 2) x + 2 + x − 4 = (x + 2) (x − 4) Ax − 4A + Bx + 2B (A + B) x + (2B − 4A) . == (x + 2) (x − 4) (x + 2) (x − 4) 6 (A + B) x + (2B − 4A) (x + 2) (x − 4) = (x + 2) (x − 4) . 6 = (A + B) x + (2B − 4A) .

0.3. Ðàöèîíàë ôóíêöèéí èíòåãðàë 46 A+B =0 , ⇒ A = −B 2B − 4A = 6 2B + 4B = 6 ⇒ A = −B , ⇒ A = −1 . 6B = 6 B=1 (x + 6 − 4) = x 1 4 − x 1 . 2) (x − + 2 Æèøýý 73. 3x (x + 1) (x − 2). 3x A B = + . (x + 1) (x − 2) x + 1 x − 2 A B A (x − 2) + B (x + 1) x + 1 + x − 2 = (x + 1) (x − 2) Ax − 2A + Bx + B (A + B) x + (B − 2A) = (x + 1) (x − 2) = (x + 1) (x − 2) . 3x (A + B) x + (B − 2A) =. (x + 1) (x − 2) (x + 1) (x − 2) 3x = (A + B) x + (B − 2A) . A+B =3 , ⇒ A + 2A = 3 B − 2A = 0 B = 2A ⇒ 3A = 3 , ⇒ A=1 . B = 2A B=2 3x 1 2 . = + 2 (x + 1) (x − 2) x + 1 x −

Ãàð÷èã 47 Æèøýý 74. x+4 x2 − 7x + 10. D = (−7)2 − 4 · 10 = 9 > 0 √ 7± 9 7±3 x1,2 = = = 2, 5. 2 2 x2 − 7x + 10 = (x − 2) (x − 5) , x+4 = x+4 . 5) x2 − 7x + 10 (x − 2) (x − x+4 AB = + . x2 − 7x + 10 x − 2 x − 5 A B A (x − 5) + B (x − 2) x − 2 + x − 5 = (x − 2) (x − 5) Ax − 5A + Bx − 2B (A + B) x − 5A − 2B = (x − 2) (x − 5) = (x − 2) (x − 5) . x+4 (A + B) x − 5A − 2B (x − 2) (x − 5) = (x − 2) (x − 5) . x + 4 = (A + B) x − 5A − 2B. A+B =1 ⇒ A=1−B −5A − 2B = 4 −5 (1 − B) − 2B = 4 ⇒ A=1−B ⇒ A=1−B −5 + 5B − 2B = 4 3B = 9 ⇒ A = −2 . B=3

0.3. Ðàöèîíàë ôóíêöèéí èíòåãðàë 48 x2 x+4 10 = x 3 5 − x 2 . − 7x + − − 2 Æèøýý 75. 2x − 3 x2 + 7x + 6. D = 72 − 4 · 6 = 25 > 0, √ −7 ± 25 −7 ± 5 x1,2 = = = −1, −6; 2 2 ⇒ x2 + 7x + 6 = (x + 1) (x + 6) . 2x − 3 = 2x − 3 . 6) x2 + 7x + 6 (x + 1) (x + 2x − 3 AB = +. (x + 1) (x + 6) x + 1 x + 6 A B A (x + 6) + B (x + 1) += x+1 x+6 (x + 6) (x + 1) Ax + 6A + Bx + B (A + B) x + (6A + B) . == (x + 6) (x + 1) (x + 6) (x + 1) 2x − 3 (A + B) x + (6A + B) =, (x + 1) (x + 6) (x + 6) (x + 1) 2x − 3 = (A + B) x + (6A + B) . A+B =2 ⇒ A=2−B 6A + B = −3 6 (2 − B) + B = −3 ⇒ A=2−B ⇒ A=2−B ⇒ A = −1 . 12 − 6B + B = −3 5B = 15 B=3

Ãàð÷èã 49 2x − 3 6 = x 3 6 − x 1 . x2 + 7x + + + 1 Æèøýý 76. x−2 x2 (x + 1). x−2 A B C = + + . x2 (x + 1) x x2 x+1 A B C Ax (x + 1) + B (x + 1) + Cx2 x + x2 + x + 1 = x2 (x + 1) = Ax2 + Ax + Bx + B + Cx2 (A + C) x2 + (A + B) x + B = . x2 (x + 1) x2 (x + 1) x − 2 = (A + C) x2 + (A + B) x + B.  A+C =0  A+B =1 .  B = −2  C = −3  A=3 .  B = −2 x−2 = 3 − 2 − x 3 . x2 (x + 1) x x2 + 1 Æèøýý 77. 1 x(2x + 1)2 .