You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
âè áóäåòå âèâ÷àòè ó êóðñ³ òåî𳿠éìîâ³ðíîñò³. Âîíà ââîäèòüñÿ<br />
ÿê ïðîäóêò ³íòåãðóâàííÿ ôóíêö³¿ f(x) =e -x2 .<br />
Òå, ùî îáåðíåíà îïåðàö³ÿ º á³ëüø ñêëàäíîþ ó ïîð³âíÿíí³<br />
ç ïðÿìîþ, âèäíî ³ç òàêèõ î÷åâèäíèõ òâåðäæåíü: 1) äîâåäåííÿ<br />
íåâèíóâàòîñò³ ï³äñóäíîãî º á³ëüø ñêëàäíîþ ïðîáëåìîþ<br />
í³æ äîâåäåííÿ éîãî âèíè; 2) óñÿêèé ñòóäåíò, ÿêèé âîëî䳺<br />
³íîçåìíîþ ìîâîþ, íàïðèêëàä ³ñïàíñüêîþ, ìîæå äîâåñòè âèêëàäà÷åâ³,<br />
ùî â³í ä³éñíî íåþ âîëî䳺. Äëÿ öüîãî éîìó òðåáà<br />
ïðîñòî çàãîâîðèòè ³ñïàíñüêîþ ìîâîþ. Òåïåð óÿâ³òü ñîá³, ùî<br />
“íåîðäèíàðíèé ñòóäåíò” çàÿâëÿº âàì, øàíîâíèé ÷èòà÷ó, ùî<br />
â³í âîëî䳺 ñòàðîäàâíüîþ ìîâîþ àöòåê³â (³íê³â). Ñïðîñòóâàòè<br />
öå òâåðäæåííÿ âàì íå âäàñòüñÿ, òîìó ùî “íåîðäèíàðíèé<br />
ñòóäåíò” íà âàø ñóìí³â çàâæäè ìîæå ñêàçàòè: ÿ âîëîä³þ<br />
ñòàðîäàâíüîþ ìîâîþ àöòåê³â (³íê³â), àëå çàðàç öüîãî ðîáèòè<br />
íå áóäó.<br />
Ïåðøå òâåðäæåííÿ ïîâ’ÿçàíî ç òàêèì þðèäè÷íèì ïîíÿòòÿì,<br />
ÿê ïðåçóìïö³ÿ íåâèíóâàòîñò³. Äðóãå æ òâåðäæåííÿ<br />
º ô³ëîñîôñüêèì ³ äåÿêîþ ì³ðîþ æàðò³âëèâèì.<br />
8.2. ÂËÀÑÒÈÂÎÑÒ² ÍÅÂÈÇÍÀ×ÅÍÈÕ ²ÍÒÅÃ-<br />
ÐÀ˲ ÒÀ ¯ÕÍß ÎÑÍÎÂÍÀ ÒÀÁËÈÖß<br />
8.2.1. Âëàñòèâîñò³ ³ ¿õ äîâåäåííÿ<br />
1. Ïîõ³äíà íåâèçíà÷åíîãî ³íòåãðàëà äîð³âíþº ï³ä³íòåãðàëüí³é<br />
ôóíêö³¿. ijéñíî, çà îçíà÷åííÿì<br />
′ ′<br />
∫ fxdx ( ) = x + C = fx ( ). (8.2.1)<br />
( ) ( ( ) )<br />
2. Äëÿ äèôåðåíö³éîâíî¿ ôóíêö³¿ ′(x) ñïðàâåäëèâà ð³âí³ñòü<br />
∫ F′ ( x) dx= F( x) + C.<br />
(8.2.2)<br />
Ä î â å ä å í í ÿ. Íåõàé ′(x) — ïîõ³äíà ôóíêö³¿ (x). Òîä³<br />
(x) º ïåðâ³ñíîþ äëÿ ôóíêö³¿ ′(x). Îòæå, ä³éñíî ìຠì³ñöå<br />
ð³âí³ñòü (8.2.2).<br />
Çàóâàæåííÿ. гâíîñò³ (8.2.1) – (8.2.2) ìîæíà â³äïîâ³äíî<br />
çàïèñàòè ùå â òàêîìó âèãëÿä³:<br />
( )<br />
d ∫ fxdx ( ) = fxdx ( ) , (8.2.3)<br />
òîáòî äèôåðåíö³àë â³ä ³íòåãðàëà äîð³âíþº ï³ä³íòåãðàëüíîìó<br />
âèðàçó<br />
∫ dF ( ( x) ) = F( x)<br />
+ C.<br />
(8.2.4)<br />
Ôîðìóëè (8.2.3) – (8.2.4) êðàñíîìîâíî ïîêàçóþòü, ùî îïåðàö³¿<br />
³íòåãðóâàííÿ òà äèôåðåíö³þâàííÿ âçàºìíî îáåðíåí³.<br />
Öåé ôàêò çàâæäè äîçâîëÿº ðåçóëüòàò ³íòåãðóâàííÿ ïåðåâ³ðèòè<br />
äèôåðåíö³þâàííÿì.<br />
3. Ñòàëèé ìíîæíèê ìîæíà âèíîñèòè çà çíàê ³íòåãðàëà.<br />
Äëÿ ôóíêö³¿ kf(x), äå k — ñòàëà, ìàºìî<br />
∫kf( x) dx = k∫ f( x)<br />
dx . (8.2.5)<br />
Ñïðàâåäëèâ³ñòü ð³âíîñò³ (8.2.5) ëåãêî ïåðåâ³ðÿºòüñÿ<br />
′<br />
kfxdx ∫ ( ) = kx ( ) + C⇒ ( kx ( ) + C)<br />
= k′<br />
( x) = kfx ( ),<br />
òîáòî k(x) +C º ïåðâ³ñíà äëÿ kf(x).<br />
4. ²íòåãðàë â³ä ñóìè ñê³í÷åííîãî ÷èñëà ôóíêö³é, ùî ìàþòü<br />
ïåðâ³ñíó, äîð³âíþº ñóì³ ³íòåãðàë³â â³ä äîäàíê³â ôóíêö³é:<br />
( ( ) ( ) K ( ))<br />
∫ f x + f x + + f x dx =<br />
1 2<br />
= ∫f1( x) dx + ∫f2( x) dx + K + ∫fn<br />
( x)<br />
dx . (8.2.6)<br />
Ïîêàæåìî ñïðàâåäëèâ³ñòü ôîðìóëè (8.2.6) íà ïðèêëàä³<br />
äâîõ ôóíêö³é. Íåõàé<br />
∫f( xdx ) = ( x) + C, ∫f( xdx ) = ( x)<br />
+ C⇒<br />
1 1 2 2<br />
n<br />
( )<br />
⇒ f( x) = ′ ( x), f ( x) = ′<br />
( x) ⇒ f( x) + f ( x) = ( x) + ( x )<br />
′ .<br />
1 1 2 2 1 2 1 2<br />
²íòåãðóþ÷è îñòàííþ ð³âí³ñòü, îòðèìàºìî<br />
( )<br />
∫ f1( x) + f2( x) dx = 1( x) + 2( x) + C= ∫f1( x) dx+<br />
∫ f2( x)<br />
dx.<br />
6. ²íòåãðàë â³ä ë³í³éíî¿ êîìá³íàö³¿ ôóíêö³é, ùî ³íòåãðóþòüñÿ,<br />
äîð³âíþº ë³í³éí³é êîìá³íàö³¿ ³íòåãðàë³â:<br />
n<br />
n<br />
∑cf i i<br />
xdx = ∑ i i<br />
i= 1 i=<br />
1<br />
∫ ( ) c∫ f( xdx ) , çîêðåìà, ³íòåãðàë â³ä ð³çíèö³ äîð³âíþº<br />
òàê³é ñàìî ð³çíèö³ ³íòåãðàë³â.<br />
264 265
Change language