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³ îñê³ëüêè<br />
+∞<br />
∫<br />
1<br />
ðîçá³æíèé (öå ³íòåãðàë ²(λ) ïðè λ = 1), òî ðîçá³æíèì º<br />
³íòåãðàë (9.6.7) âíàñë³äîê òåîðåìè 9.6.2.<br />
9.6.2. Íåâëàñí³ ³íòåãðàëè II ðîäó<br />
Ðîçãëÿíåìî òåïåð ôóíêö³þ f(x), ÿêà âèçíà÷åíà íà ï³â³íòåðâàë³<br />
[à, b), ³ íåõàé âèêîíàíà óìîâà:<br />
x→b−0<br />
dx<br />
x<br />
( )<br />
lim f x<br />
=∞ .<br />
Òî÷êó b áóäåìî íàçèâàòè îñîáëèâîþ òî÷êîþ ôóíêö³¿ f(x).<br />
Ó ö³é òî÷ö³ ôóíêö³ÿ f(õ) ìຠâåðòèêàëüíó àñèìïòîòó<br />
(ðèñ. 9.10). Íåõàé ôóíêö³ÿ f(õ) ³íòåãðîâíà íà áóäü-ÿêîìó<br />
ñåãìåíò³ [à, b – ε], äå 0 < ε < b – à.<br />
ßêùî ãðàíèöÿ ó ïðàâ³é ÷àñòèí³ (9.6.8) ³ñíóº ³ ñê³í÷åííà,<br />
òî íåâëàñíèé ³íòåãðàë ²² ðîäó íàçèâàºòüñÿ çá³æíèì, ó ïðîòèâíîìó<br />
ðàç³ íàçèâàºòüñÿ ðîçá³æíèì.<br />
ßêùî îñîáëèâîþ òî÷êîþ ôóíêö³¿ f(x) º òî÷êà à<br />
(ðèñ. 9.11), òî<br />
b<br />
a<br />
( ) = lim ∫ ( )<br />
∫f x dx<br />
b<br />
−ε→ 0a+ε<br />
f x dx<br />
çà óìîâè, ùî ôóíêö³ÿ f(x) ³íòåãðîâíà íà ñåãìåíò³ [a + ε, b],<br />
0
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