Интеграл

0.3. Ðàöèîíàë ôóíêöèéí èíòåãðàë 50 1 A BC = + + . x(2x + 1)2 2x + 1 (2x + 1)2 x 1 = (2A + 4C) x2 + (A + B + 4C) x + C,  2A + 4C = 0  A + B + 4C = 0 .  C =1  A = −2  B = −2 .  C =1 1 1)2 = 1 − 2 1 − (2x 2 1)2 . x(2x + x 2x + + Æèøýý 78. 30x (x + 1) (x − 2) (x + 3). 30x A B C = + + . (x + 1) (x − 2) (x + 3) x + 1 x − 2 x + 3 30x = A (x − 2) (x + 3)+B (x + 1) (x + 3)+C (x + 1) (x − 2) , 30x = A x2 + x − 6 + B x2 + 4x + 3 + C x2 − x − 2 , 30x = (A + B + C) x2+(A + 4B − C) x+(−6A + 3B − 2C) .  A+B+C =0  A + 4B − C = 30 .  −6A + 3B − 2C = 0 A = 5, B = 4, C = −9.

Ãàð÷èã 51 (x + 30x + 3) = x 5 1 + x 4 2 − x 9 . 1) (x − 2) (x + − + 3 Æèøýý 79. x2 + 2 . 2x x3 − 3x2 + x3 − 3x2 + 2x = x x2 − 3x + 2 = x (x − 1) (x − 2) . x2 + 2 AB C x3 − 3x2 + 2x = x + x − 1 + x − 2. x2 + 2 = A (x − 1) (x − 2) + Bx (x − 2) + Cx (x − 1) , x2 + 2 = A x2 − 3x + 2 + B x2 − 2x + C x2 − x , x2 + 2 = (A + B + C) x2 + (−3A − 2B − C) x + 2A.  A+B+C =1  −3A − 2B − C = 0 .  2A = 2 A=1  B = −3 ,  C =3 x3 x2 + 2 2x = 1 − x 3 1 + x 3 . − 3x2 + x − − 2 Æèøýý 80. 6 x (x2 + x + 3) 6 A Bx + C = + . x (x2 +x + 3) x x2 +x +3

0.3. Ðàöèîíàë ôóíêöèéí èíòåãðàë 52 6 = (A + B) x2 + (A + C) x + 3A. 6 = (A + B) x2 + (A + C) x + 3A.  A+B =0  A+C =0 .  3A = 6 A = 2, B = −2, C = −2. x (x2 6 + 3) = 2 − 2x + 2 . +x x x2 + x + 3 Æèøýý 81. 16 (x2 + x + 2) (x − 1)2 16 Ax + B C D (x2 + x + 2) (x − 1)2 = x2 + x + 2 + x − 1 + (x − 1)2 . 16 = (Ax + B) (x − 1)2+C (x − 1) x2 + x + 2 +D x2 + x + 2 16 = (Ax + B) x2 − 2x + 1 +(Cx − C) x2 + x + 2 +Dx2 + Dx + 2D, 16 = Ax3 + Bx2 − 2Ax2 − 2Bx + Ax + B + Cx3 − C¨x¨2 ¨ + C¨x¨2 − Cx + 2Cx − 2C + Dx2 + Dx + 2D, ¨ 16 = (A + C)x3 + (−2A + B + D)x2 + (A − 2B + C + D)x + B − 2C + 2D.  A+C =0   −2A + B + D = 0  A − 2B + C + D = 0 .   B − 2C + 2D = 16 

Ãàð÷èã 53 A = 3, B = 2, C = −3, D = 4. 16 3x + 2 3 4 (x2 + x + 2) (x − 1)2 = x2 + x + 2 − x − 1 + (x − 1)2 . Æèøýý 82. x2 − 6x (x − 1) (x2 + 2x + 2) x2 − 6x A Bx + C = + . (x − 1) (x2 + 2x + 2) x − 1 x2 + 2x + 2 x2 − 6x = A x2 + 2x + 2 + (Bx + C) (x − 1) , x2 − 6x = Ax2 + 2Ax + 2A + Bx2 + Cx − Bx − C, x2 − 6x = (A + B) x2 + (2A + C − B) x + (2A − C) .  A+B =1  2A + C − B = −6 .  2A − C = 0 A = −1, B = 2, C = −2. (x − x2 − 6x + 2) = x2 2x − 2 2 − x 1 . 1) (x2 + 2x + 2x + − 1

0.3. Ðàöèîíàë ôóíêöèéí èíòåãðàë 54 0.3.2 Òîäîðõîé áóñ êîýôôèöèåíòèéí àðãà àøèãëàí èíòåã- ðàë áîäîõ Ðàöèîíàë ôóíêö ãýäãýýð áèä õî¼ð îëîí ãèø³³íòèéí õàðü- öààã îéëãîíî. °ð°°ð õýëáýë P (x) õýëáýðèéí, Q(x) P (x) áà Q(x) íü îëîí ãèø³³íò áàéäàã ôóíêöèéã ðàöèîíàë ôóíêö ãýäýã. Þóíû ò³ð³³íä, P (x) îëîí ãèø³³íòèéã Q(x)- ò õóâààõ çàìààð P (x)-èéí çýðýã Q(x)-èéí çýðãýýñ ýðñ áàãà ãýæ ³çýæ áîëíî: deg P < deg Q Îäîî àëãåáðûí ³íäñýí òåîðåì ¼ñîîð, Q(x) íü íýã ãèø³³í- ò³³äýä çàäàðíà: Q(x) = A(x − a1)(x − a2)...(x − aN ). Öààøèä A = 1 ãýæ àâàõàä àëäàõ þì áàéõã³é. Õýðýâ Q-èéí êîýôôèöèåíò³³ä áîäèò áîë, áîäèò áèø ÿçãóóðóóä íü õîñ õîñîîðîî êîìïëåêñ õîñìîã áàéõ òóë, äýýðõ çàäàðãààã Q(x) = (x − a1)...(x − am)((x − b1)2 + c21)...((x − bm )2 + c2m ) õýëáýðò áè÷èæ áîëíî. Ýíä ai, bj, ck òîîíóóä íü á³ãä áîäèò òîîíóóä. Èíãýýä õýñýã÷èëñýí áóòàðõàéã àøèãëàí, P (x) ôóíêöèéã Q(x) äàðààõ õýëáýðèéí ôóíêö³³äèéí øóãàìàí ýâë³³ëýãò çàä- ëàõ áîëîìæòîé: 11 (I) : (x − a)n , (II) : ((x − a)2 + b2)n , x (III) : ((x − a)2 + b2)n .

Ãàð÷èã 55 Õýñýã÷èëñýí áóòàðõàé ãýæ þó áàéäàã áèëýý ñàíóóëàõ ³³ä- íýýñ õýäýí æèøýý àâ÷ ³çüå. x+1 1 2 (x − 1)2 = x − 1 + (x − 1)2 x2 + 6x + 2 = (x 1 − x 1 1 + x 2 2 (x + 1)2(x − 2) + 1)2 + − x3 2)2 = x 2 − 2x (x2 + x2 + (x2 + 2)2 4x3 + 3x − 1 = x+1 − 1 + 3 x2(x2 + 1) x2 + 1 x2 x P (x) = F (x) + R (x) , Q (x) Q (x) 1. Adx = A ln |ax + b| ax + b Adx A 2. (ax + b)k = a (1 − k) (ax + b)k−1 ax2 + bx + c = a b 2 4ac − b2 x+ + . 2a 4a2 Ax + B At + B 1 At + B (ax2 + bx + c)k dx = [a (t2 + m2)]k dt = ak (t2 + m2)k dt b m2 = 4ac − b2 B = B − Ab t=x+ , 4a2 , . 2a 2a 3. tdt 1 t2 + m2 t2 + m2 = 2 ln

0.3. Ðàöèîíàë ôóíêöèéí èíòåãðàë 56 dt 1 t 4. t2 + m2 = m arctg m tdt 1 5. (t2 + m2)k = 2 (1 − k) (t2 + m2)k−1 dt t 6. (t2 + m2)k = 2m2 (k − 1) (t2 + m2)k−1 2k − 3 dt + 2m2 (k − 1) (t2 + m2)k−1 Æèøýý 83. x+2 dx èíòåãðàëûã áîä. x − 1 x+2 = 1 + 3 . 1 x − 1 x − x+2 = 3 dx + 3 dx dx 1 + x − 1 dx = x−1 x − 1 = x + 3 ln |x − 1| + C. Æèøýý 84. 2x + 3 dx èíòåãðàëûã áîä. x2 − 9 2x + 3 = 2x + 3 AB = + . x2 −9 (x − 3) (x + 3) x −3 x+ 3 A (x + 3) + B (x − 3) = 2x + 3, ⇒ Ax + 3A + Bx − 3B = 2x + 3, ⇒ (A + B) x + 3A − 3B = 2x + 3.

Ãàð÷èã 57 A+B =2 , ⇒ A = 3 . 3A − 3B = 3 2 1 B = 2 2x + 3 = x 3 3 + x 1 . x2 − 9 2 2 3 − + 2x + 3 3 dx 1 dx dx = x−3 + 2 x2 − 9 2 x+3 = 3 ln |x − 3| + 1 ln |x + 3| + C = 1 (x − 3)3 (x + 3) + C. ln 22 2 Æèøýý 85. 2x2 dx èíòåãðàëûã áîä. x+1 x2 =x−1+ 1 . x+1 x+1 2x2 x2 x − 1 + 1 dx dx = 2 dx = 2 x+1 x+1 x+1 dx x2 = 2 xdx − dx + = 2 − x + ln |x + 1| + C x+1 2 = x2 − 2x + 2 ln |x + 1| + C. Æèøýý 86. x2 − 2 dx èíòåãðàëûã áîä. x+1 x2 − 2 =x−1− 1 . x+1 x+1

0.3. Ðàöèîíàë ôóíêöèéí èíòåãðàë 58 x2 − 2 x − 1 − 1 dx dx = x+1 x+1 = xdx − dx − dx = x2 − x − ln |x + 1| + C. x+1 2 Æèøýý 87. dx (2x − 1) (x + 3) èíòåãðàëûã áîä. 1 AB . = + 3 (2x − 1) (x + 3) 2x − 1 x + 1 = A (x + 3) + B (2x − 1) , 1 = Ax + 3A + 2Bx − B, 1 = (A + 2B) x + (3A − B) . A + 2B = 0 , ⇒ A + 2 (3A − 1) = 0 3A − B = 1 B = 3A − 1 ⇒ 7A − 2 = 0 ⇒ A = 2 . B = 3A − 1 7 1 B = − 7 (2x − 1 + 3) = 7 2 1) − 7 1 . 1) (x (2x − (x + 3) I= dx 2 dx − 1 dx (2x − 1) (x + 3) = 7 2x − 1 7 . x+3 I = 2 · 1 ln |2x − 1| − 1 ln |x + 3| + C 72 7 = 1 (ln |2x − 1| − ln |x + 3|) + C = 1 ln 2x − 1 + C. 7 7 x+3

Ãàð÷èã 59 Æèøýý 88. x dx èíòåãðàëûã áîä. 9 x2 − x x AB = = + . x2 − 9 (x − 3) (x + 3) x − 3 x + 3 x = A (x + 3) + B (x − 3) , x = Ax + 3A + Bx − 3B, x = (A + B) x + (3A − 3B) . A+B =1 , ⇒ A+B =1 , ⇒ A = 1 . 3A − 3B = 0 A−B =0 2 1 B = 2 x 11 1 x2 − 9 = 2 x − 3 + x + 3 . x 1 dx 1 dx dx = x−3 + 2 x2 − 9 2 x+3 = 1 ln |x − 3| + 1 ln |x + 3| + C = 1 ln |(x − 3) (x + 3)| + C 22 2 = 1 x2 − 9 + C. ln 2 Æèøýý 89. dx (x + 2)2 + 4 dx x2 + 4x + 8 èíòåãðàëûã áîä. dx dx x2 + 4x + 8 = x2 + 4x + 4 + 4 = d (x + 2) 1 x+2 = (x + 2)2 + 22 = 2 arctg 2 + C.

0.3. Ðàöèîíàë ôóíêöèéí èíòåãðàë 60 Æèøýý 90. xdx x2 + 2x + 1 èíòåãðàëûã áîä. xdx xdx x + 1 − 1 I = x2 + 2x + 1 = (x + 1)2 = (x + 1)2 dx = x+1 dx − dx dx − dx (x + 1)2 (x + 1)2 = x+1 (x + 1)2 . dx dx 1 I= − (x + 1)2 = ln |x + 1| + x + 1 + C. x+1 Æèøýý 91. x+1 (x − 2)3 dx èíòåãðàëûã áîä. x+1 x−2+3 x−2 3 (x − 2)3 dx = (x − 2)3 dx = (x − 2)3 dx+ (x − 2)3 dx = dx (x dx = − 1 +3 · 1 − 2)2 + (x − 2)2 +3 − 2)3 x − 2 (−2) (x C = − 1 2 − 2(x 3 2)2 + C x − − −2 (x − 2) − 3 −2x + 4 − 3 1 − 2x = 2(x − 2)2 + C = 2(x − 2)2 + C = 2(x − 2)2 + C. Æèøýý 92. xdx x2 + 2x + 2 èíòåãðàëûã áîä.

Ãàð÷èã 61 x2 + 2x + 2 = 2x + 2, xdx 1 2xdx 1 2x + 2 − 2 x2 + 2x + 2 = 2 x2 + 2x + 2 = 2 x2 + 2x + 2dx 1 x2 2x + 2 dx − 1 2 = + 2x + 2 2 x2 + 2x + 2dx 2 1 2x + 2 dx = x2 + 2x + 2dx − x2 + 2x + 2 = I1 − I2. 2 u = x2 + 2x + 2, du = (2x + 2) dx. 1 2x + 2 dx = 1 du = 1 ln |u| I1 = 2 2 u2 x2 + 2x + 2 1 x2 + 2x + 2 1 x2 + 2x + 2 . = ln = ln 22 dx dx dx I2 = x2 + 2x + 2 = (x2 + 2x + 1) + 1 = (x + 1)2 + 1 = arctg (x + 1) . 1 x2 + 2x + 2 − arctg (x + 1) + C. I = ln 2 Æèøýý 93. x2dx (x − 1) (x − 2) (x − 3) èíòåãðàëûã áîä. x2dx ABC . = + + 3 (x − 1) (x − 2) (x − 3) x − 1 x − 2 x − A (x − 2) (x − 3)+B (x − 1) (x − 3)+C (x − 1) (x − 2) = x2 Ax2 − 2Ax − 3Ax + 6A + Bx2 − Bx − 3Bx + 3B

0.3. Ðàöèîíàë ôóíêöèéí èíòåãðàë 62 + Cx2 − Cx − 2Cx + 2C = x2, (A + B + C) x2− (5A + 4B + 3C) x+6A + 3B + 2C = x2.  A+B+C =1  A = 1   2 5A + 4B + 3C = 0 , ⇒ B = −4 .  6A + 3B + 2C = 0  C = 9 2 x2dx = 1 − 4 + 9 2 2. (x − 1) (x − 2) (x − 3) x − 1 x − 2 x − 3 x2dx = (x − 1) (x − 2) (x − 3) 1 dx − 4 dx 9 dx = x−1 x−2 + 2 x−3 2 = 1 ln |x − 1| − 4 ln |x − 2| + 9 ln |x − 3| + C. 22 Æèøýý 94. xdx (x + 2)3 èíòåãðàëûã áîä. x A BC = + + . (x + 2)3 (x + 2)3 (x + 2)2 x + 2 x = A + B (x + 2) + C(x + 2)2, x = A + Bx + 2B + Cx2 + 4Cx + 4C, x = Cx2 + (B + 4C)x + (A + 2B + 4C).  C =0  C =0   , ⇒ B=1 . B + 4C = 1  A + 2B + 4C = 0  A = −2

Ãàð÷èã 63 x (−2) 1 (x + 2)3 = (x + 2)3 + (x + 2)2 , xdx (−2) dx dx (x + 2)3 = (x + 2)3 + (x + 2)2 = (−2) · (−2) 1 + 2)2 − x 1 2 + C = (x 1 2)2 − x 1 2 + C (x + + + 1−x−2 x+1 = + C = − + C. (x + 2)2 (x + 2)2 Æèøýý 95. dx (x + 1) (x2 + 1) èíòåãðàëûã áîä. 1 A Bx + C (x + 1) (x2 + 1) = x + 1 + x2 + 1 . A x2 + 1 + (Bx + C) (x + 1) = 1, ⇒ Ax2 + A + Bx2 + Cx + Bx + C = 1, ⇒ (A + B) x2 + (B + C) x + A + C = 1.  A+B =0  A = 1   2 B+C =0 , ⇒ B = − 1 . 2 1  A+C =1  C = 2 1 = 1 + − 1 x + 1 1) (x2 2 2 2 (x + + 1) x + 1 x2 + 1 = 1 1) − 1 · x 1 + 1 · 1 . 2 (x + 2 x2 + 2 x2 + 1

0.3. Ðàöèîíàë ôóíêöèéí èíòåãðàë 64 dx 1 dx − 1 xdx 1 dx (x + 1) (x2 + 1) = 2 x+1 2 x2 + 1 + 2 x2 + 1 = 1 ln |x + 1| − 1 d x2 + 1 1 24 + arctg x x2 + 1 2 = 1 ln |x + 1| − 1 ln x2 + 1 1 24 + arctg x + C. 2 Æèøýý 96. dx x2 (x + 1) èíòåãðàëûã áîä. 1 AB C = + + . x2 (x + 1) x2 x x + 1 1 = A (x + 1) + Bx (x + 1) + Cx2, 1 = Ax + A + Bx2 + Bx + Cx2, 1 = (B + C) x2 + (A + B) x + A.  B+C =0 A=1  A + B = 0 , ⇒ B = −1 . A=1  C =1 x2 1 1) = 1 − 1 + x 1 . (x + x2 x + 1 dx 1 − 1 + x 1 1 dx = dx − dx x2 (x + 1) = x2 x + x2 + x dx = − 1 − ln |x|+ln |x + 1| + C = ln x + 1 − 1 + C. x+1 x xx Æèøýý 97.

Ãàð÷èã 65 dx x3 + 1 èíòåãðàëûã áîä. x3 + 1 = (x + 1) x2 − x + 1 1 1 A Bx + C = = + . x3 + 1 (x + 1) (x2 − x + 1) x + 1 x2 − x + 1 A x2 − x + 1 + (Bx + C) (x + 1) = 1, ⇒ Ax2 − Ax + A + Bx2 + Cx + Bx + C = 1, ⇒ (A + B) x2 + (−A + B + C) x + A + C = 1.  A+B =0  A = 1   3 −A + B + C = 0 , ⇒ B = − 1 . 3 2  A+C =1  C = 3 1 = 1 + − 1 x + 2 = 1 − 1 · x−2 . x3 + 3 3 3 3 (x + 3 x2 − x + 1 1 x + 1 x2 − x + 1 1) dx 1 dx − 1 x−2 dx x3 + 1 = 3 x+1 3 1 x2 − x + = 1 ln |x + 1| − 1 x − 1 − 3 dx 33 x2 − 2 2 x+ 1 = 1 ln |x + 1| − 1 x2 x− 1 1 dx + 1 dx 33 −x 2 2 x2 − x + 1 + = 1 ln |x + 1| − 1 (2x − 1) dx 1 dx 36 + x2 − x + 1 2 x − 1 2 + 3 = 1 ln |x + 1| − 1 2 4 36 d x2 − x + 1 1 d x − 1 + 2 2 √2 x2 − x + 1 x − 1 2+ 3 2 2

0.3. Ðàöèîíàë ôóíêöèéí èíòåãðàë 66 = 1 ln |x + 1| − 1 ln x2 − x + 1 + √1 arctg 2x√− 1 + C. 36 33 Æèøýý 98. dx x4 − 1 èíòåãðàëûã áîä. x4 − 1 = x2 − 1 x2 + 1 = (x − 1) · (x + 1) · x2 + 1 . 1 1 A B Cx + D x4 − 1 = (x − 1) (x + 1) (x2 + 1) = x − 1 + x + 1 + x2 + 1 . A x2 + 1 (x + 1)+B x2 + 1 (x − 1)+(Cx + D) x2 − 1 = 1, Ax3 + Ax + Ax2+A + Bx3 − Bx2+Bx − B + Cx3 + Dx2 − Cx − D = 1, (A + B + C) x3+ (A − B + D) x2+ (A + B − C) x + A − B − D = 1.  A+B+C =0  A = 1   4 1  A − B + D = 0  B = − 4   A+B−C =0 , ⇒ C =0 .    A − B − D = 1  D = − 1   2 1 1 = x 1 1 − x 1 1 − x2 1 . x4 − 4 4 2 1 − + + dx 1 dx − 1 dx − 1 dx x4 − 1 = 4 x−1 4 x+1 2 x2 + 1 = 1 ln |x − 1| − 1 ln |x + 1| − 1 arctg x + C 442

Ãàð÷èã 67 1 x−1 − 1 arctg x + C. = ln 4 x+1 2 Æèøýý 99. 5x (x − 1)3 dx èíòåãðàëûã áîä. 5x A B C = + + . (x − 1)3 (x − 1)3 (x − 1)2 x − 1 A + B (x − 1) + C(x − 1)2 = 5x, ⇒ A + Bx − B + Cx2 − 2Cx + C = 5x, ⇒ Cx2 + (B − 2C) x + A − B + C = 5x.  C =0 A=5   , ⇒ B=5 . B − 2C = 5  A−B+C =0  C =0 5x 5 5 (x − 1)3 = (x − 1)3 + (x − 1)2 . 5x 55 (x − 1)3 dx = (x − 1)3 + (x − 1)2 dx =5 dx dx = 5 · (x − 1)−2 − 5 + C (x − 1)3 + 5 − 1)2 −2 − (x x 1 = − 5 1)2 − x 5 1 + C. 2(x − − Æèøýý 100. dx (x2 − 1)2 èíòåãðàëûã áîä.

0.3. Ðàöèîíàë ôóíêöèéí èíòåãðàë 68 11 (x2 − 1)2 = (x − 1)2(x + 1)2 = ABCD + + + . (x − 1)2 x − 1 (x + 1)2 x + 1 1 = A(x + 1)2 + B (x − 1) (x + 1)2 + C(x − 1)2 + D (x + 1) (x − 1)2, 1 = A x2 + 2x + 1 +(Bx − B) x2 + 2x + 1 +C x2 − 2x + 1 + (Dx + D) x2 − 2x + 1 , 1 = Ax2 + 2Ax + A + Bx3 − Bx2 + 2Bx2 − 2Bx + Bx − B + Cx2 − 2Cx + C + Dx3 + Dx2 − 2Dx2 − 2Dx + Dx + D, 1 = (B + D)x3 + (A + B + C − D)x2 + (2A − B − 2C − D)x + (A − B + C + D).  B+D=0  A = 1   2 1  A + B + C − D = 0  B = − 2   C = D = 2A − B − 2C − D = 0 , ⇒  1 .    2 1  A − B + C + D = 1  2 (x2 1 1)2 = 2(x 1 1)2 − 1 1) + 2(x 1 1)2 + 1 . − − 2 (x − + 2 (x + 1) dx dx − dx dx (x2 − 1)2 = 2(x − 1)2 2 (x − 1) + 2(x + 1)2 dx 1 dx 1 dx 1 dx + = (x − 1)2 − 2 x−1 + 2 2 (x + 1) 2 (x + 1)2 1 dx = − 1 1) − 1 ln |x − 1| − 2 1 1) + x+1 2 (x − 2 (x + 2

Ãàð÷èã 69 + 1 ln |x + 1| + C = 1 ln x+1 −1 11 +C 22 x−1 2 x−1 + x+1 1 x+1 − 1 · 2x 1 +C = 1 ln x+1 − x2 x 1 + C. = ln x−1 2 x2 − 2 x−1 − 2 Æèøýý 101. èíòåãðàëûã áîä. dx (x2 + x − 1)2 dx dx (x2 + x − 1)2 = x2 + x + 1 + 3 2 4 4 = dx 2. x + 1 2+ √2 2 3 2 (t2 dt = 2m2 (k t + m2)k−1 + 2k − 3 · + m2)k − 1) (t2 2m2 (k − 1) dt (t2 + m2)k−1 dx 2= 1 2+ √2 · 3 · · 1 2 √ 2 3 4 2 3 x + 1 2 2 1 x + 2 2 4−3 · dx 2 + 1 = 3 (x2 + x + 1) 2 · 3 · √2 4 x + 1 2+ 3 2 2

0.3. Ðàöèîíàë ôóíêöèéí èíòåãðàë 70 2 dx + 3 x + 1 2+ √2 2 3 2 = 2 + 2 · √2 arctg x+ 1 + C +x 3 3 2 √ 3 (x2 + 1) 3 2 = 2 + √4 arctg 2x√+ 1 + C. +x 33 3 3 (x2 + 1)

Ãàð÷èã 71 0.4 Èððàöèîíàë ôóíêöèéí èíòåãðàë 0.4.1 n ax + b ôóíêöèéí èíòåãðàë cx + d √Æèøýý 102. x+9 dx èíòåãðàëûã áîä. x 1 ⇒ x + 9 = u2, ⇒ x = u2 − 9, dx = 2udu. u = (x + 9)2 , √ x+9 u u2 x dx = u2 − 9 · 2udu = 2 u2 − 9du u2 − 9 + 9 9 = 2 u2 − 9 du = 2 1 + u2 − 9 du =2 du + 18 du = 2u + 18 · 1 ln u−3 +C u2 − 32 6 u+3 √ √ √x + 9 − 3 + C. = 2 x + 9 + 3 ln x+9+3 Æèøýý 103. dx√ èíòåãðàëûã áîä. x− x √ u = x1 = x, ⇒ x = u2, dx = 2udu, 2 x dx√ = 2udu &udu = 2 du −x u2 − u = 2 &u (u − 1) u−1 √ = 2 ln |u − 1| + C = 2 ln x − 1 + C. Æèøýý 104.

0.4. Èððàöèîíàë ôóíêöèéí èíòåãðàë 72 √ dx èíòåãðàëûã áîä. x+1 √ u = x, ⇒ x = u2, dx = 2udu. √ dx = 2udu u u+1−1 du =2 du = 2 x+1 u+1 u+1 u+1 = 2 1 − 1 du = 2u − 2 ln |u + 1| + C u+1 √√ = 2 x − 2 ln x + 1 + C. Æèøýý 105. √ 3 5x − 1dx èíòåãðàëûã áîä. u = (5x − 1 = √ − 1, ⇒ 5x − 1 = u3, ⇒ 5x = u3 + 1, 3 5x 1) 3 ⇒x= u3 + 1 dx = 3u2du , . 55 √ u · 3u2du = 3 u3du = 3 · u4 + C 3 5x − 1dx = 55 54 3u4 3 3 (5x − 1)4 + C. = +C = 20 20 Æèøýý 106. √ x dx èíòåãðàëûã áîä. x+1 1√ u = (x + 1)2 = x + 1. x + 1 = u2, ⇒ x = u2 − 1, dx = 2udu.

Ãàð÷èã 73 √x dx = u2 − 1 u2 − 1 du · 2udu = 2 x+1 u = 2u3 − 2u + C = 2 (x + 1)3 √ − 2 x + 1 + C. 33 Æèøýý 107. √ x 2x − 3dx èíòåãðàëûã áîä. 1√ u = (2x − 3)2 = 2x − 3, 2x + 3 = u2, ⇒ 2x = u2 + 3, ⇒ x = u2 + 3 dx = udu. , 2 √ u2 + 3 · u · udu = 1 u4 + 3u2 du x 2x − 3dx = 22 1 u5 u3 u5 u3 = +3· +C = + +C 25 3 10 2 (2x − 3)5 (2x − 3)3 = + + C. 10 2 Æèøýý 108. √dx èíòåãðàëûã áîä. x x−4 1√ u = (x − 4)2 = x − 4, ⇒ x − 4 = u2, ⇒ x = 4 + u2, dx = 2udu. √dx = 2&udu u = 2 du du x x−4 (4 + u2) 4 + u2 = 2 22 + u2 &

0.4. Èððàöèîíàë ôóíêöèéí èíòåãðàë 74 √ = 2 · 1 arctg u + C = arctg u + C = arctg x − 4 + C. 22 2 2 √Æèøýý 109. √x − 1 dx èíòåãðàëûã áîä. x+1 √ x = u, ⇒ x = u2, dx = 2udu. √ √x − 1 u−1 u2 − u I= dx = 2udu = 2 du. x+1 u+1 u+1 u2 − u =u−2+ 2 . u+1 u+1 I =2 u − 2 + 2 du = 2 udu − 4 du + 4 du u+1 u+1 = 2u2 − 4u + 4 ln |u + 1| + C = x − √ + 4 ln √ + C. 4x x+1 2 Æèøýý 110. dx èíòåãðàëûã áîä. √ x + 3x I= dx√ = dx . x+ 3x x + x1 3 ÕÁÅÕ(1, 3) = 3 u = x 1 , ⇒ x = u3, dx = 3u2du. 3 I= dx = 3u2du u2du udu 1 =3 u3 + u = 3 u2 + 1. x + x 1 u3 + (u3)3 3

Ãàð÷èã 75 t = u2 + 1, dt = 2udu, ⇒ udu = dt . 2 I =3 udu dt = 3· 1 dt = 3 ln |t| + C u2 + 1 = 3 2 t2 t2 3 u2 + 1 3 12 = ln + C = ln 2 x3 + 1 + C 2 √ 3 23 3 x2 + 1 + C. = ln x3 + 1 + C = ln 2 2 Æèøýý 111. √ dx √ èíòåãðàëûã áîä. x+ 3x I= √ dx√ = dx . x+ 3x x1 + x1 2 3 ÕÁÅÕ (2, 3) = 6. u = x 1 , ⇒ x = u6, dx = 6u5du, 6 6u5du u5du u5du I= 1 1 =6 u3 + u2 = 6 (u6)2 + (u6)3 u2 (u + 1) u3du =6 . u+1 u3 = u2 − u + 1 − 1 . u+1 u+1 I =6 u3du =6 u2 − u + 1 − 1 dx u+1 u+1

0.4. Èððàöèîíàë ôóíêöèéí èíòåãðàë 76 u3 u2 = 6 − + u − ln |u + 1| + C 32 = 2u3 − 3u2 + 6u − 6 ln |u + 1| + C. √√√ √ I = 2 x − 3 3 x + 6 6 x − 6 ln | 6 x + 1| + C. Æèøýý 112. dx èíòåãðàëûã áîä. √ 3x + 1 √ 3 x, u = x1 = ⇒ x = u3, dx = 3u2du. 3 I= √ dx = 3u2du u2 =3 du. 3x+1 u+1 u+1 u2 =u−1+ 1 , u+1 u+1 I =3 u−1+ 1 3u2 du = − 3u + 3 ln |u + 1| + C u+1 2 √ 3 3 x2 √ √ = − 3 3 x + 3 ln 3 x + 1 + C. 2 Æèøýý 113. dx èíòåãðàëûã áîä. √ 5x − 1 √ dx = dx . 5x−1 1 x1 − 5 x1 = u, ⇒ x = u5, dx = 5u4du. 5 dx 5u4du u4du I= = u − 1 = 5 u − 1. x1 − 1 5

Ãàð÷èã 77 u4 1 = u3 + u2 + u + 1 + u 1 . u− − 1 I =5 (u3 + u2 + u + 1 + u 1 )du − 1 u4 u3 u2 = 5( + + + u + ln |u − 1|) + C √4 3 √ 2 √ 5 x4 5 x3 5 x2 √ √ = 5( + + + 5 x + ln 5 x − 1 ) + C. 432 Æèøýý 114. √ dx √ èíòåãðàëûã áîä. 3x 4x − I= √ dx√ = dx . 3x− 4x x1 − x1 3 4 ÕÁÅÕ (3, 4) = 12 u = x 1 , ⇒ x = u12, dx = 12u11du. 12 12u11du 12u11du u8du I= =1 1 u4 − u3 = 12 u − 1. (u12) 3 − (u12) 4 u8 1 = u7 + u6 + u5 + u4 + u3 + u2 + u + 1 + u 1 . u− − 1 I = 12 u7 + u6 + u5 + u4 + u3 + u2 + u + 1 + u 1 1 du − = 12 u8 + u7 + u6 + u5 + u4 + u3 u2 + u + ln |u − 1| +C + 8 7 65 4 32

0.4. Èððàöèîíàë ôóíêöèéí èíòåãðàë 78 = 3u8 + 12u7 + 2u6 + 12u5 + 3u4 + 4u3 + 6u2 27 5 + 12u + 12 ln |u − 1| + C = 3 1 8 12 1 7 + x 12 x 12 27 1 6 12 1 5 14 13 12 1 +2 x 12 + x 12 +3 x 12 + 4 x 12 +6 x 12 + 12x 12 5 +12 ln 1 − 1 3 3 12 7 1 12 5 1 1 + C = x2 + x12 + 2x2 + x12 + 3x3 + 4x4 x 12 27 5 11 1 − 1 + C = 3 √ + 12 √ + √ 3 x2 x12 7 2x +6x6 + 12x12 +12 ln x 12 27 12 √ + √ + √√ + 12 √ √ + x12 5 33x 4 4 x+6 6 x 12 x+12 ln 12 x − 1 + C. 5 Æèøýý 115. √ dx √ èíòåãðàëûã áîä. 3 x2 − x I= √ dx √ = dx . 3 x2 − x x2 − x1 3 2 ÕÁÅÕ (3, 2) = 6, √ u = x1 = 6 x, ⇒ x = u6, dx = 6u5du. 6 6u5du u5du u5du I= 2 1 =6 u4 − u3 = 6 (u6)3 − (u6)2 u3 (u − 1) u2du =6 . u−1 u2 1 =u+1+ . u−1 u−1

Ãàð÷èã 79 1 u2 I =6 u+1+ du = 6 + u + ln |u − 1| + C u−1 2 √ √ √ = 3u2 + 6u+6 ln |u − 1| + C = 33x + 6 6 x+6 ln 6x−1 + C. Æèøýý 116. dx √ èíòåãðàëûã áîä. x−2 √ √ x − 2 = u2, ⇒ x = u2 + 2, ⇒ x = u2 + 2 2 = u4 + 4u2 + 4, ⇒ dx = 4u3 + 8u du. I= dx 4u3 + 8u du u2 + 2 du √= =4 x−2 u 4u3 4 √ 3+8 √ = + 8u + C = x−2 x − 2 + C. 33 Æèøýý 117. √ ex + 1 dx èíòåãðàëûã áîä. ex + 1 = u2, ⇒ exdx = 2udu, ⇒ dx = 2udu 2udu = . ex u2 − 1 I= √ 2udu u2du ex + 1dx = uu2 − 1 = 2 u2 − 1 u2 − 1 + 1 1 = 2 u2 − 1 du = 2 1 + u2 − 1 du =2 du − 2 1 du = 2u − 2 · 1 ln 1+u +C − u2 2 1−u

0.4. Èððàöèîíàë ôóíêöèéí èíòåãðàë 80 = 2u − ln 1+u √ √ + C. + C = 2 ex + 1 − ln 1 + √ex + 1 1−u 1 − ex + 1 Æèøýý 118. √ e xdx èíòåãðàëûã áîä. √ ⇒ x = t2, dx = 2tdt, t = x, √ tetdt. I = e xdx = et · 2tdt = 2 u=t  I =2 tetdt =  u = 1  = 2 tet − etdt = 2 (tet − et) + C  v = et    v=e = 2 (t − 1) et + C = 2 √ − 1) √ + C. (x ex

Ãàð÷èã 81 0.4.2 Weierstrass îðëóóëàõ sin x = 2 tg x = 2t cos x = 1−tg2 x = 1−t2 tg x = 2 tg x = 2t 2 1+t2 2 1+t2 2 1−t2 x x 1+tg2 2 1+tg2 x 1−tg2 2 2 1−tg2 x 1+tg2 x 1+tg2 x ctg x = 2 = 1−t2 sec x = 2 = 1+t2 csc x = 2 = 1+t2 x 2t x 1−t2 x 2t 2 tg 2 1−tg2 2 2 tg 2 x t = tg , x = arctg t. 2 2dt dx = d (2 arctg t) = 1 + t2 . cos2x = 1 1 = 1 1 t2 , sin2x = 1 tg2x = 1 t2 + tg2x + + tg2x + t2 Æèøýý 119. dx èíòåãðàëûã áîä. 1 + sin x x ⇒ x = 2 arctg t, 2dt t = tg , dx = 1 + t2 . 2 2t sin x = 1 + t2 , dx 2dt dt dt = 1 + t2 + 2t = (t + 1)2 1+t2 = 1 + sin x 1 + 2t 1+t2 − 22 = t + C = − + C. + 1 tg x + 1 2 Æèøýý 120. dx 3 − 2 sin x èíòåãðàëûã áîä.

0.4. Èððàöèîíàë ôóíêöèéí èíòåãðàë 82 2t 2dt x = arctg t, sin x = 1 + t2 , dx = 1 + t2 , dx 2dt 2dt 3 − 2 sin x = 3 + 3t2 − 4t I= 1+t2 = 3 − 2 · 2t 1+t2 = 2dt 2 dt . = 1 3 t2 − 4 t + 1 3 t2 − 4 t + 3 3 t2 − 4 + 1 = t2 − 4 + 2 2 22 t t +1 3 33 − 3 = t−2 2−4+1= t− 2 25 + 39 39 = t− 2 2 √2 5 + . 33 u = t− 2 du = dt , 3 2 dt 2 du I= 2=3 2+ √ √2 3 − 2 5 u2 + 5 t 3 3 3 = 2 · 1 u + C = √2 3 t− 2 +C arctg √ arctg √ 3 √ 35 5 5 5 33 = √2 arctg 3t√− 2 + C = √2 arctg 3 tg√x2 − 2 + C. 55 5 5 Æèøýý 121.

Ãàð÷èã 83 dx èíòåãðàëûã áîä. 1 + cos x 2 x ⇒d x 2dt ⇒ cos x = 1 − t2 t = tg , 2 = 1 + t2 , 2 1 + t2 . 4 dx d x 2dt 2 = = 2 1+t2 1 + cos x 1 + cos x 1 + 1−t2 2 2 1+t2 =4 dt x 1 + t2 + 1 − t2 = 2 dt = 2t + C = 2 tg + C.  4 Æèøýý 122. dx èíòåãðàëûã áîä. 1 + cos 2x dt 1 − t2 t = tg x, ⇒ x = arctg t, ⇒ dx = 1 + t2 , ⇒ cos 2x = 1 + t2 , dx dt dt = 1 + t2 + 1 − t2 1+t2 = 1 + cos 2x  1 + 1−t2 1+t2 dt t 1 = = + C = tg x + C. 22 2 Æèøýý 123. dx èíòåãðàëûã áîä. 4 + 5 cos x 2 x ⇒ x = 4 arctg t, 4dt x 1 − t2 t = tg , dx = 1 + t2 , cos = 1 + t2 . 4 2 dx 4dt 4dt 4 (1 + t2) + 5 (1 − t2) = 1+t2 = 4 + 5 cos x 4 + 5 · 1−t2 2 1+t2

0.4. Èððàöèîíàë ôóíêöèéí èíòåãðàë 84 dt = 4 4 + 4t2 + 5 − 5t2 =4 dt = 4 · 1 ln 3+t 2 3 + tg x + C. 32 − 2·3 3−t + C = ln 4 t2 3 3 − tg x 4 Æèøýý 124. dx èíòåãðàëûã áîä. sin x + cos x x ⇒ x = 2 arctg t, 2dt t = tg , dx = 1 + t2 . 2 2t 1 − t2 sin x = 1 + t2 , cos x = 1 + t2 , dx 2dt 2dt = 2t + 1 − t2 1+t2 = sin x + cos x 2t + 1−t2 1+t2 1+t2 dt dt = 2 1 − (t2 − 2t) = 2 1 − (t2 − 2t + 1 − 1) dt √ d (t − 1) = 2 2 − (t − 1)2 = 2 2 2 − (t − 1)2 √ √ − 1 + tg x = 2· √1 ln √2 + (t − 1) + C = √1 ln √2 2 22 2 + C. 2 − (t − 1) 2 + 1 − tg x 2 Æèøýý 125. dx èíòåãðàëûã áîä. sin x + cos x + 1 x 2dt 2t t = tg , ⇒ x = 2 arctg t, dx = 1 + t2 , sin x = 1 + t2 , 2

Ãàð÷èã 85 1 − t2 cos x = 1 + t2 . dx 2dt 2dt = 1+t2 1+t2 = sin x + cos x + 1 2t+1−t2+1+t2 2t + 1−t2 +1 1+t2 1+t2 1+t2 = 2dt dt = ln |t + 1| + C = ln x + C. = tg + 1 2t + 2 t+1 2 Æèøýý 126. dx èíòåãðàëûã áîä. sec x + 1 dx dx cos xdx I= = = . sec x + 1 1 +1 1 + cos x cos x x 2dt t = tg , ⇒ x = 2 arctg t, dx = 1 + t2 . 2 cos xdx 1−t2 · 2dt 1−t2 dt 1 − t2 = 1+t2 1+t2 (1+t2)2 1 + t2 dt I= =2 = 1 + cos x 1 + 1−t2 1+t2+1−t2 1+t2 1+t2 =− 1 + t2 − 2 = − 1dt + 2 dt 1 + t2 dt 1 + t2 = −t + 2 arctg t + C = − tg x + 2 arctg x +C tg 22 = x − tg x + C. 2 Æèøýý 127. dx èíòåãðàëûã áîä. 1 + csc x

0.4. Èððàöèîíàë ôóíêöèéí èíòåãðàë 86 x 2dt 1 + t2 t = tg , ⇒ x = 2 arctg t, ⇒ dx = 1 + t2 , ⇒ csc x = . 2 2t dx 2dt 2dt = 1+t2 I= 1+t2 = 1 + csc x 2t+1+t2 1 + 1+t2 2t 2t 2dt 2t 4t = 1 + t2 · (t + 1)2 = (1 + t2) (t + 1)2 dt. 4t At + B C D (1 + t2) (t + 1)2 = 1 + t2 + t + 1 + (t + 1)2 , 4t = (At + B) (t + 1)2 + C (t + 1) 1 + t2 + D 1 + t2 , 4t = (At + B) t2 + 2t + 1 +C t + 1 + t3 + t2 +D 1 + t2 , 4t = At3 + Bt2 +2At2 + 2Bt+At + B +Ct + C +Ct3 + Ct2 + D + Dt2, 4t = (A + C) t3 + (2A + B + C + D) t2 + (A + 2B + C) + (B + C + D) .  A+C =0 A=0    2A + B + C + D = 0  B = 2   C = 0 A + 2B + C = 4 , ⇒ .    B + C + D = 0  D = −2   4t 2 2 (1 + t2) (t + 1)2 = 1 + t2 − (t + 1)2 , 4t 1 2 t2 − (t 2 dt I = (1 + t2) (t + 1)2 dt = + + 1)2

Ãàð÷èã 87 =2 1 dt − 2 dt 2 + t2 (t + 1)2 = 2 arctg t + t + 1 + C = 2 arctg x + 2 + C = 2 · x + 2 + C tg 2 tg x + 1 tg x + 1 2 2 2 = x + 2 + C. tg x + 1 2 Æèøýý 128. dx sin4x + cos4x èíòåãðàëûã áîä. sin2x + cos2x = 1, sin2x + cos2x 2 = sin4x + 2 sin2x cos2x + cos4x = 1. sin4x + cos4x = 1 − 2 sin2x cos2x = 1 − (2 sin x cos x)2 2 =1− sin22x , 2 dx dx d (2x) I= sin4x + cos4x = = 2 − sin22x. 1− sin22x 2 t = tg x, ⇒ d (2x) = 1 dt + t2 . sin22x = 1 tg22x = 1 t2 + tg22x + t2 dt dt dt 2 + 2t2 − t2 = I= 1+t2 = √ 2 2 − t2 2 + t2 1+t2

0.4. Èððàöèîíàë ôóíêöèéí èíòåãðàë 88 = √1 arctg √t + C = √1 arctg tg√2x + C. 22 2 2 Æèøýý 129. dx a sin x + b cos x èíòåãðàëûã áîä. x 2dt t = tg ,⇒ x = 2 arctg t, dx = 1 + t2 . 2 2t 1 − t2 sin x = 1 + t2 , cos x = 1 + t2 , dx 2dt 2dt = 2at + b − bt2 1+t2 = a sin x + b cos x 2at + b−bt2 1+t2 1+t2 2 dt 2 dt = = b 1− t2 − 2a t b 1− t2 − 2a t + 4a2 − 4a2 b b b2 b2 2 dt 2 d t − a = 2=b b b 1 + 4a2 − t − a 2 2 b2 b b2+4a2 − t − a b b √ b2+4a2 a =2· 1 ·ln √b + t − b +C √ b 2 b2+4a2 b2+4a2 a b − t − b b √ b2 + 4a2 + b tg x − a =√ 1 ·ln √ 2 + C. b2 + 4a2 b2 + 4a2 − b tg x + a 2 Æèøýý 130. dx èíòåãðàëûã áîä. 3 sin x + 4 cos x

Ãàð÷èã 89 x ⇒ x = 2 arctg t, 2dt 2t t = tg , dx = 1 + t2 , sin x = 1 + t2 , 2 1 − t2 cos x = 1 + t2 . dx 2dt 2dt = 6t + 4 − 4t2 1+t2 = 3 sin x + 4 cos x 3 · 2t + 4 · 1−t2 1+t2 1+t2 2 dt 1 dt = = 4 1− t2 − 6 t 2 1− t2 − 6 t + 9 − 9 4 4 16 16 1 dt 1 dt = = 2 9 − − 3 2 2 5 2− − 3 2 1 + 16 t 4 4 t 4 = 1 · 1 ln 5 + t − 3 1 1 + t +C 2 2· 4 4 + C = ln 2 5 5 − t + 3 5 2−t 4 4 4 1 1 + 2t 1 1 + 2 tg x + C. = ln 4 − 2t + C = ln 2 5 5 4 − 2 tg x 2 Æèøýý 131. dx èíòåãðàëûã áîä. 5 sin x + 12 cos x x ⇒ x = 2 arctg t, 2dt 2t t = tg , dx = 1 + t2 , sin x = 1 + t2 , 2 1 − t2 cos x = 1 + t2 , dx 2dt = 1+t2 12 sin x + 13 cos x 12 · 2t + 13 · 1−t2 1+t2 1+t2

0.4. Èððàöèîíàë ôóíêöèéí èíòåãðàë 90 2dt = 10t + 12 − 12t2 2 dt 1 dt = = 1− t2 − 10 t 6 1 + 25 − t − 5 2 12 12 144 12 1 dt = 1 · 1 ln 13 + t − 5 +C = 6 12 12 6 13 2− t − 5 2 2 · 13 13 − t + 5 12 12 12 12 12 1 4 + t 1 4 + 6t +C = ln 6 + C = ln 9 − 6t 13 9 − t 13 6 1 4 + 6 tg x + C. = ln 2 13 9 − 6 tg x 2 Æèøýý 132. dx 5 sin x + 2 cos x + 2 èíòåãðàëûã áîä. x ⇒ x = 2 arctg t, 2dt 2t t = tg , dx = 1 + t2 , sin x = 1 + t2 , 2 1 − t2 cos x = 1 + t2 . dx 2dt = 1+t2 5 sin x + 2 cos x + 2 5 · 2t + 2 · 1−t2 + 2 1+t2 1+t2 = 2dt dt = 1 ln |5t + 2| + C 10t + 2 − 2t2 + 2 + 2t2 = 5t + 2 5 1x = ln 5 tg + 2 + C. 52 Æèøýý 133.

Ãàð÷èã 91 dx 2 sin x − cos x + 5 èíòåãðàëûã áîä. x ⇒ x = 2 arctg t, ⇒ dx = 2dt t = tg , 1 + t2 , 2 2t 1 − t2 sin x = 1 + t2 , cos x = 1 + t2 . dx 2dt 2 sin x − cos x + 5 = I= 1+t2 2 · 2t − 1−t2 + 5 1+t2 1+t2 2dt 2dt dt 6t2 + 4t + 4 = 3t2 + 2t + 2. = 1+t2 = 4t−1+t2+5+5t2 1+t2 I= dt 1 dt 3t2 + 2t + 2 = 3 t2 + 2 t + 2 3 3 1 dt 1 dt = = 3 t2 + 2 t + 1 − 1 + 2 3 1 2 5 3 9 9 3 t + 3 + 9 1 dt = 1 · 1 arctg t + 1 +C = 3 3 √ √ √ 3 5 2 5 5 1 2+ 3 t + 3 33 = √1 arctg 3t√+ 1 + C = √1 arctg 3 tg√x2 + 1 + C. 55 5 5 Æèøýý 134. 2dx 4 sin x − 3 cos x + 5 èíòåãðàëûã áîä. x ⇒ x = 2 arctg t, 2dt 2t t = tg , dx = 1 + t2 , sin x = 1 + t2 , 2

0.4. Èððàöèîíàë ôóíêöèéí èíòåãðàë 92 1 − t2 cos x = 1 + t2 . 2dx 2 · 2dt 4 sin x − 3 cos x + 5 = 1+t2 4 · 2t − 3 · 1−t2 + 5 1+t2 1+t2 4dt 4dt = 8t − 3 (1 − t2) + 5 (1 + t2) = 8t − 3 + 3t2 + 5 + 5t2 4dt dt dt = 8t2 + 8t + 2 = 2 4t2 + 4t + 1 = 2 (2t + 1)2 =2· −1 · 2t 1 1 + C = − 1 1 + C = − tg 1 + 1 + C. 2 + 2t + 2 x 2 Æèøýý 135. dx sin x + cos x − 1 èíòåãðàëûã áîä. x ⇒ x = 2 arctg t, 2dt 2t t = tg , dx = 1 + t2 , sin x = 1 + t2 , 2 1 − t2 cos x = 1 + t2 , dx 2dt sin x + cos x − 1 = I= 1+t2 2t + 1−t2 − 1 1+t2 1+t2 2dt 2dt dt = 2t + 1 − t2 − 1 − t2 = 2t − 2t2 = t − t2 . 1 AB , = + t t (1 − t) t 1 − 1 = A (1 − t) + Bt,

Ãàð÷èã 93 1 = A − At + Bt, 1 = (B − A) t + A. B−A=0 , ⇒ A=1 . A=1 B=1 dt 11 dt − dt I = t − t2 = t + 1 − t dt = t t−1 = ln |t| − ln |t − 1| + C = ln t + C = ln tg x + C. t−1 2 tg x − 1 2 Æèøýý 136. dx 1 + tg x èíòåãðàëûã áîä. t = tg x, ⇒ x = arctg t, dt dx = 1 + t2 . I= dx dt dt = (1 + t) (1 + t2). 1+t2 = 1 + tg x 1+t 1 A Bt + C (1 + t) (1 + t2) = 1 + t + 1 + t2 . 1 = A 1 + t2 + (Bt + C) (1 + t) , ⇒ 1 = A + At2 + Bt + C + Bt2 + Ct.  A+C =1  A = 1   2 A+B =0 , ⇒ B = − 1 . 2 1  B+C =0  C = 2 1 1 1−t dt = 1 ln |t + 1| − 1 t−1 I= 1 + t + 1 + t2 22 1 + t2 dt 2

0.4. Èððàöèîíàë ôóíêöèéí èíòåãðàë 94 = 1 ln |t + 1| − 1 2tdt 1 dt 24 1 + t2 + 2 1 + t2 = 1 ln |t + 1| − 1 d 1 + t2 1 24 + arctg t 1 + t2 2 = 1 ln |t + 1| − 1 ln t2 + 1 1 + arctg t + C 24 2 = 1 ln |tg x + 1| − 1 ln tg2x + 1 x + + C. 24 2

Ãàð÷èã 95 0.5 Òðèãîíîìåòðèéí èíòåãðàë 0.5.1 cos ax cos bxdx, sin ax cos bxdx, sin ax sin bxdx, sinm x cosn xdx, tgn xdx èíòåãðàë 1. cos ax cos bxdx, sin ax cos bxdx, sin ax sin bxdx, èíòåãðàë • cos ax cos bx = 1 + bx) + cos(ax − bx)] [cos(ax 2 • sin ax cos bx = 1 + bx) + sin(ax − bx)] [sin(ax 2 • sin ax sin bx = − 1 + bx) − cos(ax − bx)] [cos(ax 2 2. sinmx cosnxdx èíòåãðàë 1 Õýðýâ m ñèíóñûí çýðýã ñîíäãîé, áàéâàë u-îðëóóëàëòûã àøèãëàíà. u = cos x, du = − sin xdx 2 Õýðýâ n êîñèíóñûí çýðýã ñîíäãîé áàéâàë u-îðëóóëàëòûã àøèãëàíà. u = sin x, du = cos xdx 3 Õýðýâ m, n çýðýã òýãø áàéâàë äàâõàð °íöãèéí òîìü¼î- ãîîð çàäàëíà. sin2 x = 1 − cos 2x cos2 x = 1 + cos 2x , 22

0.5. Òðèãîíîìåòðèéí èíòåãðàë 96 sinn xdx = −sinn−1 x cos x + n − 1 sinn−2 xdx nn cosn xdx = −cosn−1 x sin x + n − 1 cosn−2 xdx nn 3. tgn xdx èíòåãðàë 1 + tg2 x = sec2 x tgn−2x sec2x − 1 dx tgnxdx = tgn−2x tg2xdx = = tgn−1x − tgn−2xdx. ctgn−2x csc2x − 1 dx n−1 4. 1 + ctgnx = cscnx ctgnxdx = ctgn−2x ctg2xdx = ctgn−1x ctgn−2xdx. =− n−1 − 5. secn xdx èíòåãðàë secnxdx = secn−2x tg x + n − 2 secn−2xdx. n−1 n−1 6. cscn xdx èíòåãðàë cscnxdx = − cscn−2x ctg x + n − 2 cscn−2xdx. n− 1 n − 1

Ãàð÷èã 97 Æèøýý 137. sin3xdx èíòåãðàëûã áîä. u = cos x, du = − sin xdx sin3xdx = sin2x sin xdx = 1 − cos2x sin xdx = − 1 − u2 du = u2 − 1 u3 du = − u + C cos3x 3 = − cos x + C. 3 Æèøýý 138. cos5xdx èíòåãðàëûã áîä. u = sin x, du = cos xdx, cos2x = 1 − sin2x, cos5xdx = cos2x 2 cos xdx = 1 − sin2x 2 cos xdx = 1 − u2 2du = 1 − 2u2 + u4 du = u − 2u3 + u5 + C 35 = sin x − 2sin3x sin5x + + C. 35 Æèøýý 139. sin6xdx èíòåãðàëûã áîä. sin2x = 1 − cos 2x cos2x = 1 + cos 2x I= sin6xdx = , 22

0.5. Òðèãîíîìåòðèéí èíòåãðàë 98 sin2x 3dx = 1 (1 − cos 2x)3dx 8 1 (1 − 3 cos 2x + 3cos22x − cos32x dx = 8 = x − 3 · sin 2x + 3 cos22xdx − 3 cos32xdx. 88 2 8 8 cos22xdx = 1 + cos 4x 1 (1 + cos 4x) dx dx = 22 1 sin 4x x sin 4x = x+ =+ . 2 4 28 cos32xdx, u = sin 2x, du = 2 cos 2xdx. cos32xdx = 1 2cos22x cos 2xdx 2 1 2 1 − sin22x 1 1 − u2 du = u − u3 = cos 2xdx = 26 22 = sin 2x − sin32x . 26 I = x − 3 sin 2x 3 x sin 4x −1 sin 2x − sin32x + + + 8 16 8 2 8 82 6 C 5x sin 2x 3 sin 4x sin32x =− + + + C. 16 4 64 48 Æèøýý 140. sin2x cos3xdx èíòåãðàëûã áîä.

Ãàð÷èã 99 u = sin x, du = cos xdx. sin2x cos3xdx = sin2x cos2x cos xdx = sin2x 1 − sin2x cos xdx = u2 1 − u2 du = u2 − u4 du = u3 − u5 + C = sin3x − sin5x + C. 35 35 Æèøýý 141. sin2x cos4xdx èíòåãðàëûã áîä. I = sin2x cos4xdx = (sin x cos x)2cos2xdx. sin x cos x = sin 2x , cos2x = 1 + cos 2x sin2x = 1 − cos 2x , . 22 2 I= sin 2x 2 1 + cos 2x dx = 1 sin22x (1 + cos 2x) dx 22 8 1 sin22xdx + 1 sin22x cos 2xdx = 1 1 − cos 4x = dx 88 82 1 2sin22x cos 2xdx = 1 (11 cos 4x) dx + 16 16 1 sin22x d (sin 2x) = 1 x − sin 4x 1 · sin32x + C + + 16 16 4 16 3 x sin 4x sin32x =− + + C. 16 64 48 Æèøýý 142.