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8.3.<br />
⎛<br />
∫ ⎜sin x+<br />
⎝<br />
3 ⎞<br />
dx<br />
⎟dx<br />
; 8.4. ∫<br />
4<br />
;<br />
x ⎠<br />
x<br />
8.5. ∫ dx<br />
2 2<br />
cos xsin<br />
x<br />
; 8.6.<br />
2<br />
∫ ctg xdx ;<br />
− x<br />
⎛ 7 ⎞<br />
x<br />
8.7. ∫ 7 ⎜<br />
5+<br />
dx<br />
2 ⎟ .<br />
⎝ 1−<br />
x ⎠<br />
8.3.2. Ìåòîä ï³äñòàíîâêè<br />
Ñóòü öüîãî ìåòîäó ñêëàäàºòüñÿ ó ââåäåíí³ ï³ä çíàê ³íòåãðàëà<br />
òàêî¿ íîâî¿ çì³ííî¿, ï³ñëÿ ï³äñòàíîâêè ÿêî¿ âèõ³äíèé<br />
³íòåãðàë ñïðîùóºòüñÿ àáî çâîäèòüñÿ äî òàáëè÷íîãî. ϳäñòàíîâêè<br />
áóâàþòü äâîõ òèï³â:<br />
1. Ïåðøèé òèï ï³äñòàíîâêè. Íåõàé â³äîìî, ùî<br />
∫ gudu ( ) = Gu ( ) + C. (8.3.1)<br />
Çðîáèìî çàì³íó: u = ϕ(x), äå ôóíêö³ÿ ϕ(x) äèôåðåö³éîâíà<br />
â äåÿêîìó ³íòåðâàë³. Òîä³ ôîðìàëüíî ð³âí³ñòü (8.3.1) ïåðåòâîðèòüñÿ<br />
â òàêó:<br />
∫ g( ϕ( x)) ϕ ′( x) dx = G( ϕ ( x))<br />
+ C. (8.3.2)<br />
Çã³äíî ç çàóâàæåííÿì 2 ï. 8.2.1 ñïðàâåäëèâ³ñòü ð³âíîñò³<br />
(8.3.2) ïåðåâ³ðèìî øëÿõîì äèôåðåíö³þâàííÿ îáîõ ¿¿ ÷àñòèí.<br />
Ìàºìî<br />
( ∫ g( ϕ( x)<br />
) ϕ ′( x)<br />
dx) = g( ϕ( x)<br />
) ϕ′<br />
( x),<br />
′<br />
′ ′<br />
ϕ = ⋅ϕ = ϕ = ϕ ϕ .<br />
( G( ( x)<br />
)) ( G( u)<br />
) u ′( x) g( u) ′( x) g( ( x)<br />
) ′( x)<br />
Îñòàíí³ äâ³ ð³âíîñò³ ïîêàçóþòü, ùî ð³âí³ñòü (8.3.2) ïðè<br />
âèêîíàíí³ âêàçàíèõ âèùå óìîâ ä³éñíî ñïðàâåäëèâà.<br />
Ðîçãëÿíåìî äâà òåîðåòè÷íèõ ïðèêëàäè.<br />
Ïðèêëàä 8.3.1. ³äîìî, ùî<br />
Òðåáà çíàéòè ∫ fax ( + bdxa ) , ≠0.<br />
Ðîçâ’ÿçàííÿ<br />
⎡t = ax+<br />
b⎤<br />
1 1 1<br />
∫f ( ax + b) dx = ⎢<br />
f () t dt () t C ( ax b)<br />
C<br />
dt adx<br />
⎥ = ∫ = + = + + .<br />
⎣ = ⎦ a a a<br />
Îòæå,<br />
1<br />
∫ f ( ax + b ) dx = ( ax + b) + C, a ≠ 0 . (8.3.4)<br />
a<br />
Ðîçãëÿíåìî òåïåð êîíêðåòèçîâàí³ ïðèêëàäè.<br />
Ïðèêëàä 8.3.2. Çíàéòè ( ) 2001<br />
∫ 5x+<br />
7 dx .<br />
2001<br />
Ðîçâ’ÿçàííÿ. Ïîêëàäåìî: fx ( ) = x . Òîä³ êîðèñòóþ-<br />
÷èñü ïîäâ³éíèìè ôîðìóëàìè (8.3.3) – (8.3.4), ìàòèìåìî<br />
( )<br />
( x + ) 2002<br />
2002<br />
2001 x<br />
2001 5 7<br />
∫x dx = + C, ∫ 5x+ 7 dx = + C.<br />
2002 10010<br />
Óÿâ³òü ñîá³, øàíîâíèé ÷èòà÷ó, ÿê³ á âèíèêëè ïðîáëåìè<br />
ïðè ³íòåãðóâàíí³ çàïðîïîíîâàíîãî íåâèçíà÷åíîãî ³íòåãðàëà<br />
ÿêùî ìåòîä ï³äñòàíîâêè íå áóâ áè âàì â³äîìèé. Òðåáà áóëî<br />
äâî÷ëåí (5õ + 7) ï³äíåñòè çà ôîðìóëîþ á³íîìà Íüþòîíà â<br />
2001-é ñòåï³íü. Ïðè öüîìó îäåðæàòè 2002 äîäàíêè, à ïîò³ì<br />
ïðî³íòåãðóâàòè. Òèòàí³÷íà ïðàöÿ! ßê âè çðîçóì³ëè, íå çàâæäè<br />
âîíà ïîòð³áíà.<br />
Çàóâàæèìî òàêîæ, ùî ïðè â³äïîâ³äíîìó òðåíóâàíí³ ÷èòà÷<br />
çìîæå çàïèñóâàòè ðåçóëüòàò ³íòåãðóâàííÿ ³íòåãðàë³â âèäó<br />
∫ fax ( + bdxa ) , ≠ 0 çðàçó. Ïåâíà ð³÷, ùî ïðè öüîìó òðåáà çíàòè<br />
òàáëè÷í³ ³íòåãðàëè.<br />
Ïðèêëàä 8.3.3. Çíàéòè ³íòåãðàë<br />
Ðîçâ’ÿçàííÿ<br />
∫<br />
sin( 7x + 5) dx .<br />
1<br />
∫ sin(7x+ 5) dx=− cos(7x+ 5) + C .<br />
7<br />
∫ fxdx ( ) = x ( ) + C. (8.3.3)<br />
268 269
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