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9.4. ÎÑÍÎÂÍÀ ÔÎÐÌÓËÀ ²ÍÒÅÃÐÀËÜÍÎÃÎ<br />
×ÈÑËÅÍÍß<br />
9.4.1. ²íòåãðàë ³ç çì³ííîþ âåðõíüîþ ìåæåþ<br />
Íåõàé ôóíêö³ÿ f(õ) íåïåðåðâíà íà ñåãìåíò³ [à, b], òîä³<br />
âîíà ³íòåãðîâíà íà íüîìó. ³çüìåìî äîâ³ëüíå õ∈[à, b], òîä³<br />
f(õ) áóäå ³íòåãðîâíîþ íà [a, õ], òîáòî ³ñíóº ³íòåãðàë<br />
x<br />
a<br />
()<br />
∫ f t dt.<br />
Òóò ìè çì³ííó ³íòåãðóâàííÿ õ çàì³íèëè íà t, ùîá íå<br />
ïëóòàòè ³ç çì³ííîþ âåðõíüîþ ìåæåþ õ.<br />
ßêùî õ çì³íþºòüñÿ, òî, î÷åâèäíî, áóäå çì³íþâàòèñü ³ öåé<br />
³íòåãðàë, òîáòî â³í ÿâëÿºòüñÿ ôóíêö³ºþ çì³ííî¿ õ. Ïîçíà÷èìî<br />
öþ ôóíêö³þ Ô(õ):<br />
x<br />
()<br />
Ô( x ) = ∫ f t dt. (9.4.1)<br />
a<br />
²íòåãðàë (9.4.1) íàçèâàºòüñÿ ³íòåãðàëîì ³ç çì³ííîþ âåðõíüîþ<br />
ìåæåþ.<br />
Òåîðåìà 9.4.1 (Áàððîó 1 ). Íåõàé ó ð³âíîñò³ (9.4.1)<br />
ôóíêö³ÿ f(õ) íåïåðåðâíà íà ñåãìåíò³ [à, b]. Òîä³ ñïðàâåäëèâà<br />
ôîðìóëà<br />
′<br />
x<br />
Ô( ′ )<br />
⎛<br />
f<br />
⎞<br />
x = ⎜∫<br />
() t dt⎟<br />
= f( x ). (9.4.2)<br />
⎝ a ⎠<br />
Ä î â å ä å í í ÿ. Çàñòîñóºìî îçíà÷åííÿ ïîõ³äíî¿ äëÿ ôóíêö³¿.<br />
Òîä³ (äèâ. ï. 7.2.1) ìàòèìåìî<br />
x+∆x x<br />
f<br />
( ) ( ) () −<br />
x x x ∫ t dt ∫f()<br />
Φ +∆ −Φ<br />
t dt<br />
a<br />
a<br />
Φ ′( x)<br />
= lim<br />
= lim<br />
=<br />
∆x→0 ∆x<br />
∆x→0<br />
∆x<br />
1<br />
x+∆x<br />
= lim ⋅ ∫ f() t dt = limf() ξ = limf() ξ = f( x)<br />
. (9.4.3)<br />
∆x→0∆x<br />
x<br />
∆x→0<br />
ξ→x<br />
1<br />
Áàððîó ²ñààê (1630 – 1677) — àíãë³éñüêèé ìàòåìàòèê, ô³çèê, ô³ëîñîô,<br />
áîòàí³ê ³ òåîëîã, â÷èòåëü ñàìîãî Íüþòîíà.<br />
Ó ëàíöþæêó ð³âíîñòåé (9.4.3) ìè ïîñë³äîâíî çàñòîñóâàëè<br />
âëàñòèâ³ñòü 4 (ï. 9.3), òåîðåìó 9.3.1 ïðî ñåðåäíº çíà÷åííÿ ³<br />
íåïåðåðâí³ñòü ôóíêö³¿ f(õ) (çà òåîðåìîþ ïðî ñåðåäíº çíà÷åííÿ<br />
òî÷êà ξ çíàõîäèòüñÿ ì³æ òî÷êàìè x i x + ∆x, ³ òîìó ïðè<br />
∆x<br />
→0<br />
ξ→ x ). Îòæå, òåîðåìó äîâåäåíî, îñê³ëüêè ñàìà ë³âà<br />
÷àñòèíà â ð³âíîñò³ (9.4.2) äîð³âíþº ñàì³é ïðàâ³é ¿¿ ÷àñòèí³.<br />
Òåîðåìà Áàððîó ìຠäóæå âàæëèâå çíà÷åííÿ. Âîíà ñòâåðäæóº<br />
³ñíóâàííÿ ïåðâ³ñíî¿ ó íåïåðåðâíî¿ ôóíêö³¿ ³ âñòàíîâëþº<br />
çâ’ÿçîê ì³æ íåâèçíà÷åíèì ³íòåãðàëîì òà ³íòåãðàëîì ç³<br />
çì³ííîþ âåðõíüîþ ìåæåþ. Êð³ì òîãî, ó â³äïîâ³äíîñò³ äî<br />
ð³âíîñò³ (9.4.2) ôóíêö³ÿ Ô(õ) º ïåðâ³ñíîþ äëÿ ôóíêö³¿ f(õ).<br />
9.4.2. Ôîðìóëà Íüþòîíà — Ëåéáí³öà<br />
Íåõàé (õ) — áóäü-ÿêà ïåðâ³ñíà ôóíêö³ÿ äëÿ íåïåðåðâíî¿<br />
ôóíêö³¿ f(x) íà ñåãìåíò³ [à, b]. Îñê³ëüêè Ô(õ) òåæ<br />
ïåðâ³ñíà äëÿ f(x), òî çã³äíî ç îñíîâíîþ ëåìîþ ³íòåãðàëüíîãî<br />
÷èñëåííÿ ìàòèìåìî ð³âí³ñòü<br />
x<br />
a<br />
() = ( ) + ,<br />
∫ f t dt x C<br />
ÿêà ñïðàâåäëèâà äëÿ áóäü-ÿêîãî õ∈[a, b]. Ïîêëàäåìî õ = à.<br />
Îñê³ëüêè<br />
a<br />
∫<br />
a<br />
çâ³äêè Ñ =–(à), òîáòî<br />
() ( )<br />
f t dt = 0, òî a + C = 0,<br />
x<br />
a<br />
() = ( ) − ( ).<br />
∫ f t dt x a<br />
Òåïåð ïîêëàäåìî òóò õ = b. ijñòàíåìî:<br />
b<br />
a<br />
() = () − ( ),<br />
∫ f t dt b a<br />
àáî çã³äíî ç âëàñòèâ³ñòþ 1 (ï. 9.3)<br />
b<br />
a<br />
( ) = ( ) − ( )<br />
∫ f x dx b a . (9.4.4)<br />
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