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14.3. 14.5. α+ 1 α x ∫ xdx= + C , α≠−1. 14.4. α+ ∫ cos xdx = sin x + C . 1 ∫ = ∫ dx = ln + x −1 x dx x C x x a x x 14.7. ∫adx= + C; ∫edx= e + C. ln a 14.8. 2 ∫ cosec xdx =− ctgx + C . dx 1 x 14.9. ∫ 2 2 = arctg + C . x + a a a dx 1 x − a 14.10. ∫ = 2 2 ln + C . x − a 2a x+ a 2 . 14.6. ∫sec xdx = tg x + C . ìåòîä ï³äñòàíîâêè — ( ) = ( ) ( = const) ∫af x dx a∫ f x dx a ; () = ( ϕ( )) ϕ ′( ) ( = ϕ( )) ∫f t dt ∫ f x x dx t x ; ³íòåãðóâàííÿ ÷àñòèíàìè — ( ) ( ) ( ) ′( ) ( ) ( ) ( ) ′( ) ∫uxdvx = ∫uxv xdx= uxvx− ∫vxu xdx= ( ) ( ) vxdux ( ) ( ) = uxvx−∫ . 14.4.8. Âèçíà÷åíèé ³íòåãðàë ³ éîãî âëàñòèâîñò³: b b ∫ f( x) dx = ( x) = ( b) − ( a) , ′ ( x) = f( x) ; a a b a β ( ) = ( ϕ( )) ϕ ′() ( = ϕ() , ϕ( α ) = , ϕ() β = ) ∫f x dx ∫ f t t dt x t a b ; α 14.11. ∫ dx 2 2 x ± a 2 2 = + ± + ln x x a C . b b b ∫uxdvx ( ) ( ) = uxvx ( ) ( ) −∫ vxdux ( ) ( ); a a a dx 14.12. ∫ 2 2 a − x x = arcsin + C a . dx ⎛ x π ⎞ dx x 14.13. ∫ = ln tg⎜ + ⎟ + C . 14.14. ∫ = ln tg + C . cos x ⎝2 4 ⎠ sin x 2 14.4.7. Îñíîâí³ âëàñòèâîñò³ ³ ïðàâèëà ³íòåãðóâàííÿ: ′ ∫ ∫ ; ( f( x) dx) = f( x) ; f′ ( x) dx = f( x) + C ( C = const) b a a ( ) ( ) ( ) ( ) ∫f x dx =− ∫f x dx a < b ; ∫ f x dx = 0; a b a b b b ( ) ± ( ) ⎤ = ( ) ± ( ) ∫⎡⎣f x g x ⎦ dx ∫f x dx ∫g x dx; a a a b a b ( ) = ( ) ( = ) ∫kf x dx k∫ f x dx k const ; b c b ( ) = ( ) + ( ) a a c a ∫f x dx ∫f x dx ∫ f x dx , a < c < b. ( ) ± ( ) ⎤ = ( ) ± ( ) ∫⎡⎣f x g x ⎦ dx ∫f x dx ∫g x dx; 506 507
14.5. ÃÐÀÔ²ÊÈ ÄÅßÊÈÕ ÎÑÍÎÂÍÈÕ ÅËÅÌÅÍÒÀÐÍÈÕ ÔÓÍÊÖ²É 14.5.1. y = kx + b, k =tgα (ðèñ. 14.8). Ðèñ. 14.10 14.5.4. y = k x (ðèñ. 14.11). Ðèñ. 14.8 14.5.2. y = a x , a >0, a ≠ 1 (ðèñ. 14.9). Ðèñ. 14.11 Ðèñ. 14.9 14.5.5. y = log a x (ðèñ. 14.12). 1 14.5.3. y = x n , y = x n , n ∈ N. Íà ðèñ. 14.10 ö³ ôóíêö³¿ çîáðàæåíî ïðè n = 2 òà n =3. Ðèñ. 14.12 508 509
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Ì²Í²ÑÒÅÐÑÒÂÎ ÎÑÂ²Ò
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2.4. Ðàíã ìàòðèö³ ......
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7.9. Ïîõ³äí³ âèùèõ ïî
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10.2. Ãðàíèöÿ ³ íåïåð
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Ïåðåäìîâà Ç óñ³õ ñî
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äåòüñÿ” âèêîðèñòî
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1 îòðèìàºìî äð³á m = 2
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1.2.7. Ìîäóëü ä³éñíîã
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Òåìà 2 Îñíîâè àëãåá
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Öÿ ñèñòåìà ð³âíÿíü
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2 0 . Ó ÿêîñò³ åëåìåí
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Äîâ³ëüíèé åëåìåíò
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Ïðèêëàä 2.2.2. Çíàéòè
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a a ... a a ... a a a ... a a ... a
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Òîä³ a 12 12 ∆ 1 =∆+ m =∆
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Äâ³ ìàòðèö³ À ³ Â íà
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3 0 . Îá÷èñëèòè àëãåá
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ÒÅÌÀ 3 ÑÈÑÒÅÌÈ Ë²Í²
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Ïðè ðîçâ’ÿçàíí³ ñè
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äå æa11 a12 ... a ö 1n a21 a22
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Ðîçâ’ÿçàííÿ. Øëÿõî
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3.3. Êîðèñòóþ÷èñü ïð
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Äëÿ äàíî¿ òî÷êè Ì â
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Îòæå, êîîðäèíàòè òî
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ìèì îáìåæåííÿì, îáó
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Ïðèêëàä 4.2.6. Ñêëàñò
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Ïðèêëàä 4.2.14. Çíàéòè
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ð³âíÿííÿìè ïåðøîãî
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ßêùî k çàäàíå ÷èñëî,
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3) ∆ =0 ³ ∆ x = 0, ∆ y = 0.
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ÿêèé íàðîäèâñÿ â ëþ
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º àíàë³òè÷íå ð³âíÿ
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4.5. ÏÎÍßÒÒß ÏÐÎ Ð²ÂÍ
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Ç ðîçãëÿíóòîãî ïðè
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⎧1 ⎫ Ïðèêëàä 5.1.10. Ï
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5.1.5. Îñíîâí³ òåîðåì
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Îòðèìàëè, ùî ð³çíèö
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Â òàê³é çàãàëüí³é ï
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n n 1 ⎛⎛1+ 5⎞ ⎛1− 5⎞
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1. Àíàë³òè÷íèé ñïîñ
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òâ³, ïðè òåõí³÷íèõ
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4. Äëÿ ïîáóäîâè ãðàô
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Ðîçâ’ÿçàííÿ. Íåõàé
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Öÿ ôóíêö³ÿ ÿâëÿº ñî
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Çã³äíî ç îçíà÷åííÿ
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sin x- 0 < x- 0
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Îçíà÷åííÿ 6.3.5 Ôóíêö
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6.3.6. Ãëîáàëüí³ âëàñ
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Ñïðàâåäëèâ³ñòü ôîð
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6.23. Îá÷èñëèòè ãðàíè
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Çà ïåð³îä ÷àñó â³ä t
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Ïðèêëàä 7.3.2. Ïîêàçà
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Â ÿêîñò³ ïðèêëàäà ð
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Òåîðåìà 7.6.2 (ïðî äèô
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Òàáëèöÿ ïîõ³äíèõ ñ
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7.13. r(ϕ) =ϕ sin ϕ + cos ϕ; î
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7.9.8. Çíàõîäèìî y′: Ò
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Ôóíêö³ÿ ïðîïîçèö³¿
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Îñê³ëüêè â êðèòè÷í
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ïðè ∆x =1 i x=1000: (∆y)⏐x
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Íàñë³äîê 3. ßêùî íà
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7.14.9. = ln x 1/ x = 1/ x 1 − =
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Çàóâàæåííÿ. Ðîçêëà
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Îö³íèìî n ( ) n+ 1 θx n+ 1
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Çâ³äñè ó êð - ó äîò
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Ð î ç â ’ ÿ ç à í í ÿ.
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âè áóäåòå âèâ÷àòè ó
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8.3. ⎛ ∫ ⎜sin x+ ⎝ 3 ⎞ dx
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8.12. ∫ sin( ax + b) dx ; 8.13.
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Ïåðøèé òèï äîäàíê³
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8.5. ²ÍÒÅÃÐÓÂÀÍÍß ÄÅ
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6 4 1 1 1 5 1 3 = ∫t − dt+ ∫t
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[ñ, b] ñåãìåíòà [a, b] ¿
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9.2.1. Îçíà÷åííÿ òà óì
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4. ßêùî ôóíêö³ÿ ó = f(
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9.4. ÎÑÍÎÂÍÀ ÔÎÐÌÓËÀ
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â³äíîñò³ äî ôîðìóë
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Îçíà÷åííÿ 9.6.1. Íåâë
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³ îñê³ëüêè +∞ ∫ 1 ðî
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Ïðèêëàä 9.6.12 (òåîðåò
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Ïðèêëàä 9.7.2. Îá÷èñë
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Îñê³ëüêè ôóíêö³ÿ ó
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Ãðàíèöÿ ö³º¿ ñóìè ï
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Ð î ç â ’ ÿ ç à í í ÿ.
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Òîä³ ìîæíà ïîêàçàò
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4 3 ∆Ψ = 10 − 10 = 9000 . Îò
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ïàäàº ç íàö³îíàëüí
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Çàóâàæåííÿ 2. Àíàëî
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10.2. ÃÐÀÍÈÖß ² ÍÅÏÅÐ
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Öå ð³âíÿííÿ ïðÿìî¿,
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Öåé æàðò³âëèâèé åê
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äàòêîâ³é âàðòîñò³
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Îçíà÷åííÿ 10.4.2 Äèôå
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∂u äå ( i = 1, 2, K , n) ∂x
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Àíàëîã³÷íî ( 0, 0 ) ( 0,
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Çàóâàæèìî, ùî ïîíÿò
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Òîä³: 1) ÿêùî a a a a > 0,
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ÂÏÐÀÂÈ Äîñë³äèòè í
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Òàêèì ÷èíîì, ìè âèç
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Ðèñ. 10.21 Â³äîìèé Îìó
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ßñíî, ùî âàð³àíò ë³
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Page 227 and 228: S n 1 1 1 = + + K + 1⋅2 2⋅ 3 n
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Page 233 and 234: íà çàâæäè íå ïîðîæí
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Page 239 and 240: êö³þ, êîòðó íàé÷àñò
Page 241 and 242: Âèðàç âèäó a+bi=z, äå à
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Page 245 and 246: 13.16. z =− 2+ 2 3i ; 13.17. z =
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Page 249 and 250: y ⎛ π ⎞ x tg α= k , k ; ctg (
Page 251 and 252: 1 2 tg α+ ctg α= = sin αcos α s
Page 253: 14.4. ÅËÅÌÅÍÒÈ ÂÈÙÎ¯
Page 257 and 258: 14.5.11. y = arctg x (ðèñ. 14.19
Page 259 and 260: 14.7. ÍÀÁËÈÆÅÍÅ ÇÍÀ×Å
Page 261: Ê36 Êåðåêåøà Ï. Â. Ëå
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