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Ïðèçíàê ñðàâíåíèÿ. Ïóñòü f (x) è g(x) èíòåãðèðóåìû íà êàæäîì [a; β] è g(x) > f (x) > 0 íà [A; +∞) ïðè íåêîòîðîì A > a. ÒîãäàZ+∞g(x)dxañõîäèòñÿ =⇒Z+∞f (x)dxañõîäèòñÿ.Äîêàçàòåëüñòâî. Åñëè óñëîâèå êðèòåðèÿ Êîøè âûïîëíåíî äëÿ g(x), òî îíî âûïîëíåíî è äëÿ f (x).Ñëåäñòâèå 12.1. Ïðåäåëüíûé ïðèçíàê ñðàâíåíèÿ.Ïóñòü f (x) è g(x) èíòåãðèðóåìû íà êàæäîì [a; β],g(x) > 0, f (x) > 0 è f (x) ∼ g(x) ïðè x → +∞. Òîãäà+∞Zg(x)dxñõîäèòñÿ+∞Z⇐⇒f (x)dxañõîäèòñÿ.a(x) 1Äîêàçàòåëüñòâî.
∃N >a: fg(x)− 1 <ïðè x > N . Òîãäà ìîæíî ïðèìåíèòü2ëåììó 12.3 ê íåðàâåíñòâàìf (x) < 2g(x) è g(x) < 2f (x).Ïðèìåð 12.4.+∞Ïðîâåðèì íà ñõîäèìîñòü èíòåãðàë0ïðèçíàê ñðàâíåíèÿ. Ïðè x > 1 èìååì e−xZ2+∞Ze−x dxñõîäèòñÿ2. Èíòåãðàë íåáåðóùèéñÿ.
Ïðèìåíèìe−x dx6 e−x;+∞Z2=⇒e−x dx0053ñõîäèòñÿ.Ïðèìåð 12.5.Èññëåäóåì èíòåãðàë5Z√2dx.2x − 4 îêðåñòíîñòè îñîáîé òî÷êè 2 ïðèìåíèì ïðåäåëüíûé ïðèçíàê ñðàâíåíèÿ, âñïîìíèâýêâèâàëåíòíîñòü at − 1 ∼ t ln a ïðè t → 0. Ïðè x → 2 èìååì2x − 4 = 4(2x−2 − 1) ∼ 4 ln 2 (x − 1);1√2 ln 2Ïðèìåð 12.6.ZZ5dx√x−1ñõîäèòñÿ =⇒2Z5√dx2x − 4ñõîäèòñÿ.2+∞sin xÈíòåãðàëdx ñõîäèòñÿ óñëîâíî.x0Äåéñòâèòåëüíî, èíòåãðàë îò ìîäóëÿ ôóíêöèè ðàñõîäèòñÿ:ZπN0ZπnNNXX| sin x|| sin x|2dx >dx =>xπnπnn=1n=1π(n−1)NZZ+1N n+12Xdx2dx2 ln(N + 1)=−→ +∞.>=N →∞π n=1xπxπn1Òåïåðü ïðè β > π/2 ïðèìåíèì ôîðìóëó èíòåãðèðîâàíèÿ ïî ÷àñòÿì:Zβ+∞βZβZ−d cos x− cos x −dxdx=+cos x 2 −→ 0 −cos x 2 .xxx β→+∞xπ2π/2π/2π/2+∞cos x dxÝòîò ïðåäåë ñóùåñòâóåò è êîíå÷åí, ïîñêîëüêó íåñîáñòâåííûé èíòåãðàëx2π/2 cos x àáñîëþòíî ñõîäèòñÿ ïî ïðèçíàêó ñðàâíåíèÿ: x2 6 x−2 (ñì.
ïðèìåð 12.1). Íàîòðåçêå æå [0; π/2] ôóíêöèÿ sin x/x èíòåãðèðóåìà.ZÏðèìåð 12.7.¾Ïàðàäîêñ ìàëÿðà¿. Ïóñòü ïîëîâèíà âåòâè ãèïåðáîëû {y = 1/x, x > 1}âðàùàåòñÿ âîêðóã ñâîåé àñèìïòîòû Ox (ñì. ðèñ. íà ñ. 63). Âû÷èñëèì âíóòðåííèéîáúåì ïîëó÷àþùåéñÿ òðóáû:ZβV∗ = supβ>1π11−dx=πlim+= π < +∞.β→+∞x2β1154Bû÷èñëèì ïëîùàäü áîêîâîé ïîâåðõíîñòè:ZbS = 2π limb→+∞1p1 + x−4 dx =x1=√1 p1 + x−4 − 1 πlim 2 1 + x−4 + ln √= +∞.2 b→+∞1 + x−4 + 1 bÄîêàçàòü ðàñõîäèìîñòüýòîãî èíòåãðàëà ìîæíî è áåç òî÷íîãî âû÷èñëåíèÿ: ïðè1p1x → +∞ èìååì1 + x−4 ∼ è ïðèìåíèì ïðåäåëüíûé ïðèçíàê ñðàâíåíèÿ.xxÏåðåéäåì ê ðàññìîòðåíèþ ñîñòàâíûõ íåñîáñòâåííûõ èíòåãðàëîâ.
ÏóñòüI çàìêíóòûé ïðîìåæóòîê, a ôóíêöèÿ f (x) èìååò íà I êîíå÷íîå ìíîæåñòâîîñîáûõ òî÷åê. Ñîñòàâíîé íåñîáñòâåííûé èíòåãðàë ôóíêöèè f ïî I îïðåäåëÿåòñÿñëåäóþùèì îáðàçîì.Ðàçîáüåì I íà îòðåçêè [ak−1 ; ak ], êàæäûé èç êîòîðûõ ñîäåðæèò ëèøü îäíóîñîáóþ òî÷êó, ÿâëÿþùóþñÿ åãî êîíöîì, à òàêæå ëó÷è, åñëè:I íå îãðàíè÷åí ñëåâà, òî (−∞; a0 ] áåç îñîáûõ òî÷åê;I íå îãðàíè÷åí ñïðàâà, òî [aN ; +∞) áåç îñîáûõ òî÷åê.Ðàññìîòðèì íåñîáñòâåííûå èíòåãðàëû 2-ãî ðîäàR ïî îòðåçêàì è 1-ãî ðîäà ïî ëó÷àì.Åñëè âñå èíòåãðàëû ñõîäÿòñÿ, òî ãîâîðÿò, ÷òî f (x)dx ñõîäèòñÿ è ðàâåí èõ ñóììåI(ñïîñîá ðàçáèåíèÿ íå âëèÿåò íà ðåçóëüòàò: ïåðåäâèíóòü íåîñîáóþ òî÷êó ðàçáèåíèÿRbèç a â b îçíà÷àåò âû÷åñòü f (x)dx èç îäíîãî ñëàãàåìîãî è ïðèáàâèòü ñòîëüêî æåaRê äðóãîìó).
Åñëè îäíî èç ñëàãàåìûõ ðàñõîäèòñÿ, òî f (x)dx ðàñõîäèòñÿ.IÏðèìåð 12.8.+∞Z−∞√3dx=x + x5 Z−1Z0+−∞Z1+−1+∞Z+01√3dx.x + x5Âñå ÷åòûðå èíòåãðàëààáñîëþòíîñõîäÿòñÿ ïî ïðåäåëüíîìóïðèçíàêó ñðàâíåíèÿ:ïðè x → 0 (x + x5 )−1/3 ∼ |x|−1/3 , ïðè x → ∞ (x + x5 )−1/3 ∼ |x|−5/3 .Ïîäûíòåãðàëüíàÿ ôóíêöèÿ íå÷åòíà, ïîýòîìó èíòåãðàë îò −∞ äî +∞ ðàâåí 0.Çàäà÷è ïî òåìå ëåêöèè 12Èññëåäîâàòü íà ñõîäèìîñòü íåñîáñòâåííûå èíòåãðàëû. Åñëè ôóíêöèÿ çíàêîïåðåìåííàÿ, âûÿñíèòü, ñõîäèòñÿ èíòåãðàë àáñîëþòíî èëè óñëîâíî.Íåñîáñòâåííûå èíòåãðàëû 1-ãî ðîäà:Z12.1.0+∞x+1dx.x2 + 5Z12.2.0+∞dx.x ln (x + 2)255+∞Z12.3.0cos x√dx.
12.4.x3 + 8Z+∞πsin xdx.ln xÍåñîáñòâåííûå èíòåãðàëû 2-ãî ðîäà:1Z√312.5.01Z12.7.0sin(1/x)√.x1/2Zdx. 12.6.2x − x201Z12.8.0dx.x ln xdx.ln cos xÑîñòàâíûå íåñîáñòâåííûå èíòåãðàëû:1Z0dx.x ln2 xZπ12.9.12.11.0√Z+∞e−x√ dx.x+∞x dx.x3 − 112.10.dx. 12.12.sin x0Z1Îòâåòû ê çàäà÷àì1.1. 103 (5x + 1) + C . 1.2.
91 (x + 2) − 14 (x + 2) + C .1.3. C − e . 1.4. 2 arctg √x + C . 1.5. 14 tg x + C .1.6. 12 ln(1 + x ) − arctg x + C (äëÿ äâóõ ñëàãàåìûõ ðàçíûå çàìåíû).1.7. 31 (2 ln x + 1) + C . 1.8. x arcsin x + √1 1− x + C .1.9. x arctg x − 12 ln(1 + x ) + C .1.10. − x2 + 14 cos 2x + x2 sin 2x + C .1.11. x − 3x + 6x − 6 e + C . 1.12. tg x ln sin x − x + C .2/398−x2 /24223/222232x2.1. 12 ln(x − 2x + 10) − 13 arctg x −3 1 + C .2.2. ln |x + 1| − ln |x| − x1 + C .2.3. − 54 ln |x| + 78 ln |x − 2| + 38 ln |x + 2| + C .2.4. 121 ln |x − 2| − 241 ln(x + 2x + 4) − 4√1 3 arctg x√+31 + C .13 ln |x + 4|+ C.2.5. − x4 + 3 ln |x| + 162.6.
x2 + 2x − 12 ln(x − 2x + 5) − 112 arctg x −2 1 + C .2.7. x + ln xx −+ 11 + C .2.8. x + x −1 1 − 4 ln |x − 1| + 8 ln |x − 2| + C .22222.9. x4 − 9x2 + 812 ln(x + 9) + C .2.10. x2 + 6x + 12 ln |x − 1| − 16 ln |x − 2| + 812 ln |x − 3| + C .42223.1. C − cos x + 23 cos x − 15 cos x.
3.2. 12 cos x − ln | cos x| + C .3.3. 12 arctg sin2 x + C . 3.4. 3x8 + sin42x + sin324x + C .352573.5. x8 − sin324x + C . 3.6. tg2 x + ln | cos x| + C .2−2√+ C.3.7. 12 arctg tg2x + C . 3.8. √25 Arth tg(x/2)511√+ C . 3.10. cos x −cos 5x + C .3.9. √12 arctg tg(x/2)2102 1 + √x √ +Cln |x − 1| + ln1− x4.1. 2 x − 2 arctg x + C . 4.2.4.3. (x + 6)√√5 + 4x − x + C .
√ 4.4. (x− 1) x − 4 + 4 lnx + x − 4 + C.4.5. 18 (2x − x)√x − 1 − lnx + √x − 1 + C .4.6. 18 (2x + 5x)√x + 1√+ 3 lnx + √x + 1 + C .4.7. arcsin(x − 1) + −1 +x −2x1 − x + C .4.8. √x √+ 4x − x + 2 lnx + 2 + √x + 4x + C .4.9. 12 ln √11 ++ xx +− 11 − √1 + x + C .4.10. 281 (4x + x − 3) √x + 1 + C .√√.222323222222−4−4−463335.1. S = n2n− 1 , S = n2n+ 1 . 5.2. S = 255, S = 255.5.3. Ïðèìåíèì êðèòåðèé Äàðáó: I = 0 6= I2 = 1.5.4. Ïðèìåíèì êðèòåðèé Äàðáó: I = I = 0.5.5.
Ðàçíîñòü S − S = n1 f (b) − f (a) ìîæíî ñäåëàòü ñêîëü óãîäíî ìàëîé.∗∗∗∗∗∗∗∗∗∗6.1. 21. 6.2. 14,5. 6.3. ln 10. 6.4. 125.6.5. 1/5. 6.6. 3/8. 6.7. 1/3.7.1. 25 arcsin √25 − arcsin √15 − ln 4. 7.2. 343.6√7.3. 2π 3 − 43 . 7.4. 8π . 7.5. 34 π − √3.7.6. π/2. 7.7.
5π/32.38.1. 32/15. 8.2. 256/15. 8.3. 8. 8.4. π/12.8.5. π/3. 8.6. 2π(π − 2). 8.7. 32π/105. 8.8. 8π/3.9.1. √5 + 12 ln(2 + √5). 9.2. √17 eπ/2−158. 9.3. 8.9.4. 22/3. 9.5. 2π√1 + k . 9.6. Îáå êðèâûå èìåþò äëèíó2(èíòåãðàë íåáåðóùèéñÿ).32π/5.10.1. 2π 2√5 + ln(2 + √5)√. 10.2.√√4( 2 + 1) 10.3. π 17 − 2 + ln √17 + 1 . 10.4. 2π√2 103 − π .10.5.
2πRD.11.1. A = %gH πR /12. 11.2. 1 − (1/2) ≈ 49 ìèí.11.3. A= 9 Äæ. 11.4. J = M a /24. 11.5. C(0; 0; 2H/5).R + 2Rr + 3r.11.6. C 0; 0; 4(R+ Rr + r )225/22222212.1. Ðàñõîäèòñÿ. 12.2. Ñõîäèòñÿ.12.3. Ñõîäèòñÿ àáñîëþòíî. 12.4. Ñõîäèòñÿ óñëîâíî.12.5. Ñõîäèòñÿ. 12.6. Ðàñõîäèòñÿ.12.7. Ñõîäèòñÿ àáñîëþòíî. 12.8. Ðàñõîäèòñÿ.12.9.
Ðàñõîäèòñÿ íà ïðàâîì êîíöå. 12.10. Ñõîäèòñÿ.12.11. Ñõîäèòñÿ. 12.12. Ðàñõîäèòñÿ íà ëåâîì êîíöå.59Z02πp1 + cos2 t dt.
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