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Ðîçâ’ÿçàííÿ. Ó â³äïîâ³äíîñò³ äî ïðèêëàäó 9.6.2 ïðè<br />
λ = 1 íåâëàñíèé ³íòåãðàë (9.6.3) ðîçá³ãàºòüñÿ. À ïðè λ≠1<br />
ìàºìî:<br />
⎧ 1<br />
1 −λ+ 1 ⎪ , ÿêùî λ> 1;<br />
∫ = ∫ x dx= = ( A − 1)<br />
= ⎨λ−1<br />
−λ + 1 −λ + 1<br />
⎪<br />
⎩ +∞ , ÿêùî λ < 1.<br />
1<br />
A<br />
+∞<br />
A<br />
−λ+<br />
dx<br />
−λ x<br />
λ lim lim lim<br />
1 x A→+∞1<br />
A→+∞ A→+∞<br />
1<br />
Îòæå, ò³ëüêè ïðè λ > 1 íåâëàñíèé ³íòåãðàë I(λ) çá³ãàºòüñÿ.<br />
ßñíî, ùî ïðè ³íøèõ çíà÷åííÿõ λ â³í ðîçá³ãàºòüñÿ.<br />
Çàóâàæåííÿ 1. Àíàëîã³÷íî ìîæíà ïîêàçàòè (ðåêîìåíäóºìî<br />
öå çðîáèòè ÷èòà÷åâ³), ùî íåâëàñíèé ³íòåãðàë<br />
∫ ( a > 0<br />
+∞<br />
dx<br />
λ<br />
) çá³ãàºòüñÿ ïðè λ > 1, à ïðè λ≤1 ðîçá³ãàºòüñÿ.<br />
a x<br />
Çàóâàæåííÿ 2. Âñòàíîâëåííÿ çá³æíîñò³ ïðè îá÷èñëåíí³<br />
íåâëàñíèõ ³íòåãðàë³â º ïåðøîñòåïåíåâîþ çàäà÷åþ, îñîáëèâî,<br />
ÿêùî òî÷íî íåâëàñíèé ³íòåãðàë íå îá÷èñëþºòüñÿ. Öåé ôàêò<br />
ïîÿñíþºòüñÿ òèì, ùî ò³ëüêè çà óìîâè çá³æíîñò³ íåâëàñíèõ<br />
³íòåãðàë³â ¿õ ìîæíà îá÷èñëþâàòè (òî÷íî àáî íàáëèæåíî).<br />
³äçíà÷èìî òàêîæ, ùî çã³äíî ç îçíà÷åííÿì íåâëàñíîãî ³íòåãðàëà<br />
éîãî ìîæíà íàáëèæåíî îá÷èñëèòè ç áóäü-ÿêîþ òî÷í³ñòþ.<br />
Íà ïðàêòèö³ îïåðàö³ÿ çíàõîäæåííÿ íàáëèæåíîãî çíà-<br />
÷åííÿ íåâëàñíîãî ³íòåãðàëà çä³éñíþºòüñÿ çà äîïîìîãîþ êîìï’þòåð³â.<br />
Ó çâ’ÿçêó ç îñòàíí³ì çàóâàæåííÿì ñôîðìóëþºìî ó âèãëÿä³<br />
òåîðåì äâ³ äîñòàòí³ óìîâè çá³æíîñò³ íåâëàñíèõ ³íòåãðàë³â<br />
² ðîäó.<br />
Òåîðåìà 9.6.1. ßêùî íà ïðîì³æêó [à, +∞) ôóíêö³¿ f(õ)<br />
³ g(õ) íåïåðåðâí³ é 0 ≤ f(õ) ≤ g(õ), òî ³ç çá³æíîñò³ ³íòåãðàëà<br />
âèïëèâຠçá³æí³ñòü<br />
+∞<br />
∫ g( x)<br />
dx<br />
(9.6.4)<br />
a<br />
+∞<br />
∫ f( x)<br />
dx, (9.6.5)<br />
a<br />
à ³ç ðîçá³æíîñò³ ³íòåãðàëà (9.6.4) âèïëèâຠðîçá³æí³ñòü ³íòåãðàëà<br />
(9.6.5).<br />
Òåîðåìà 9.6.2. ßêùî ³ñíóº ãðàíèöÿ<br />
( )<br />
( )<br />
f x<br />
lim<br />
x→+∞<br />
g x<br />
= k, 0 < k < +∞,<br />
òî îáèäâà ³íòåãðàëè (9.6.4) ³ (9.6.5) àáî âîäíî÷àñ çá³ãàþòüñÿ,<br />
àáî âîäíî÷àñ ðîçá³ãàþòüñÿ.<br />
Ïðèêëàä 9.6.7. Äîñë³äèòè íà çá³æí³ñòü ³íòåãðàë<br />
+∞<br />
xdx<br />
∫<br />
1<br />
23<br />
x + 5<br />
. (9.6.6)<br />
Ðîçâ’ÿçàííÿ. Ïðè x ≥ 1 ìàºìî î÷åâèäíó îö³íêó<br />
x x 1<br />
< =<br />
x + 5 x x<br />
23 23 21<br />
2 2<br />
+∞<br />
dx<br />
³ îñê³ëüêè ³íòåãðàë ∫ 21<br />
1<br />
x 2<br />
çá³ãàºòüñÿ (öå ³íòåãðàë ²(λ) äëÿ<br />
21<br />
λ= >1), òî çã³äíî ç òåîðåìîþ 9.6.1 çá³ãàºòüñÿ é ³íòåãðàë<br />
2<br />
(9.6.6).<br />
Ïðèêëàä 9.6.8. Äîñë³äèòè íà çá³æí³ñòü ³íòåãðàë<br />
+∞<br />
Ðîçâ’ÿçàííÿ. Ìàºìî<br />
( x)<br />
1<br />
( ln ( 1+ x)<br />
−ln<br />
)<br />
,<br />
∫ x dx . (9.6.7)<br />
x<br />
ln 1+ −ln x ⎛ ⎛ 1⎞⎞<br />
⎛ 1⎞<br />
lim = lim x ln 1 lim ln 1<br />
x→+∞ 1<br />
⎜ ⋅ ⎜ + ⎟⎟= ⎜ + ⎟ =<br />
x→+∞ ⎝ x⎠ x→+∞<br />
⎝ x⎠<br />
x<br />
⎝<br />
⎠<br />
x<br />
⎛ ⎛ 1 ⎞ ⎞<br />
= ln lim 1+ = ln e = 1,<br />
⎜ ⎜ ⎟<br />
x→+∞⎝<br />
x ⎠ ⎟<br />
⎝<br />
⎠<br />
310 311
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