nesob-int (1111230), страница 7
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Äîêàçàòü, ÷òî íåñîáñòâåííûé èíòåãðàë ïåðâîãî ðîäàf (x)dx ñõîaäèòñÿ òîãäà è òîëüêî òîãäà, êîãäà ñõîäèòñÿ ðÿä∞Pn=1Zanun , ãäå un =f (x)dx,an−1êàêîâà áû íè áûëà ïîñëåäîâàòåëüíîñòü (an ):a = a0 < a1 < a2 < . . . < an < . . . , an → +∞.Z+∞2. Äîêàçàòü, ÷òî íåñîáñòâåííûé èíòåãðàë ïåðâîãî ðîäàf (x)dx, ãäåaf (x) ≥ 0 íà [a; +∞), ñõîäèòñÿ, åñëè íàéä¼òñÿ ïîñëåäîâàòåëüíîñòü (an ):a = a0 < a1 < a2 < . .
. < an < . . . , an → +∞,50Îãëàâëåíèåòàêàÿ, ÷òî ðÿä∞Pn=1Zanun , ãäå un =f (x)dx, ñõîäèòñÿ.an−13. Ïðîâåðèòü íåïðåðûâíîñòü íà âåùåñòâåííîé îñè ôóíêöèé:Z1Z2x2 dx2à) F (y) = sin(x y)dx; á) F (y) =.1 + x2 + x4 y 20−14. Ïóñòü f íåïðåðûâíà è íåîòðèöàòåëüíà íà [0; 1], ïðè÷¼ì f (x) íå ðàâZ1af (x)dxíà òîæäåñòâåííî íóëþ. Äîêàçàòü, ÷òî ôóíêöèÿ F (a) =2x + a20òåðïèò ðàçðûâ â òî÷êå a = 0.5. Ìîæíî ëè ñîâåðøèòü ïðåäåëüíûé ïåðåõîä ïîä çíàêîì èíòåãðàëà,Z1x − xy22âû÷èñëÿÿ lime dx?y→0y206.
Äîêàçàòü âîçìîæíîñòü ïðåäåëüíîãî ïåðåõîäà ïîä çíàêîì èíòåãðàëඵZ3Zπ/2xy1x2â âûðàæåíèÿõ: à) lim arctgdx; á) limsin dx.y→oy→+∞x+yxy−107. Âû÷èñëèòü:Z1Z2Z2dxln(x + |α|)y −x2 yà) lim; á) limdx; â) limedx;n22n→∞n→∞y→+∞1 + (1 + x/n)ln(x + α )x+y01Z1Z3ã) limy→0sin(xy)dx; ä) limy→0(x + y)y + 10µarctgxyx+y1¶dx.−18.
Ìîæíî ëè èçìåíèòü ïîðÿäîê èíòåãðèðîâàíèÿ â èíòåãðàëàõ:Zπ/4 Z1Z1 Z1tg(xy)x−yà)dy pdx; á) dydx?22(x + y)3x +y +1−π/4000Z1ln(x2 +a2 )dx9. Ìîæíî ëè âû÷èñëèòü ïðîèçâîäíóþ ôóíêöèè F (a) =0ïðè a = 0, äèôôåðåíöèðóÿ ïî ïàðàìåòðó ïîä çíàêîì èíòåãðàëà?10. Èññëåäîâàòü âîçìîæíîñòü äèôôåðåíöèðîâàíèÿ ïî ïàðàìåòðó ïîäZ11dxçíàêîì èíòåãðàëà â èíòåãðàëå F (y) = cos 2 · 2.x x + |x| + 2−111. Íàéòè F 0 (y), åñëè:Z3Z3dxcos(x3 y)dx; á) F (y) = ch(x2 y 4 ) · .à) F (y) =xx112. Íàéòè F 0 (y), åñëè:23. Èíòåãðàëû, çàâèñÿùèå îò ïàðàìåòðàZ2yà) F (y) =Zy2sin(xy)dx; á) F (y) =xy512ex y dx;3yZyeysin yZln(1 + x2 y 2 )dx; ã) F (y) =â) F (y) =sh(x2 y)dx.cos yye−y113. Ïóñòü f ∈ C(R). Äîêàçàòü, ÷òî F (y) =2aZaf (x + y)dx: à) íåïðå−aðûâíà íà R; á) äèôôåðåíöèðóåìà íà R.14.
Ïóñòü f ∈ C(Π), Π = [a; b] × [c; d], à g ∈ R[a; b]. Òîãäà ôóíêöèÿZbF (y) = f (x, y)g(x)dx íåïðåðûâíà íà [c; d].a15. Ïóñòü f ∈ C(Π), Π = [a; b] × [c; d], à g ∈ R[a; b]. Òîãäà ôóíêöèÿZbF (y) =f (x, y)g(x)dx èíòåãðèðóåìà íà [c; d] è ñïðàâåäëèâî ðàâåíñòâîaZbZbF (y)dy =aZdf (x, y)dx.g(x)dxac∂f16. Ïóñòü f,∈ C(Π), Π = [a; b]×[c; d], à g ∈ R[a; b]. Òîãäà ôóíêöèÿ∂yZbF (y) = f (x, y)g(x)dx íåïðåðûâíî äèôôåðåíöèðóåìà íà [c; d].a17. Âû÷èñëèòü:ZπZπ/2à) I(a) = ln(1 − 2a cos x + a2 )dx; á)I(a) =ln(a2 cos2 x + sin2 x)dx.0018. Èññëåäîâàòü íà ðàâíîìåðíóþ ñõîäèìîñòü íà óêàçàííûõ ìíîæåñòâàõ ñëåäóþùèå èíòåãðàëû:Z+∞ p 2Z+∞ln (x + 1)sin(xy)√ dx (0 ≤ y < +∞); á)√à)dx (0, 1 ≤ p ≤ 10);x xx x−110Z+∞â)dx(1 < α0 ≤ α < +∞); ã)x lnα x2Z+∞Z+∞α −2xã)x e dx (1 ≤ α ≤ 3); ä)1Z1å)0Z1/2dx(1 < α0 ≤ α < +∞);x |ln x|α0xdx(−∞ < α ≤ α0 < 0);1 + (x − α)40xα arctg(αx)√dx (−2 < α < +∞);1 − x252ÎãëàâëåíèåZ+∞æ)x2 − α 2dx (α ∈ R); ç)(x2 + α2 )21Z+∞0ln(1 + xα )p√ dxx+ xµ¶1−∞ < α <.219.
Äîêàçàòü ðàâíîìåðíóþ ñõîäèìîñòü íà óêàçàííûõ ìíîæåñòâàõ ñëåäóþùèõ èíòåãðàëîâ:Z+∞ln x√ cos(αx)dx (< α0 ≤ α < +∞);à)x2Z+∞αá)sin(2x) sin dx (0 ≤ α ≤ 1);x0Z+∞â)ln2 xsin(3x)dx (1 < α0 ≤ α < +∞);(x − 1)22Z+∞ã)sin(αx5 )dx (0 < α0 ≤ α < +∞);x0Z+∞Z+∞cos(αx2 )dx (1 ≤ α < +∞); å)ä)0Z+∞æ)1sin(x2 )dx (0 ≤ α < +∞);1 + xα1sin(α2 x)√arctg(αx)dx (|α| ≥ 1); ç)3x2Z+∞sin(ex )dx (0 ≤ α < +∞).1 + xα020.
Èññëåäîâàòü íà ðàâíîìåðíóþ ñõîäèìîñòü íà óêàçàííûõ ìíîæåñòâàõ ñëåäóþùèå èíòåãðàëû:Z+∞Z+∞dxln(ex − x)à)(1<α<+∞);á)dx (2 < α < +∞);1 + xαxα00Z+∞â)sin xy·arctg(xy)dx (0 < y < +∞);xy+11Z+∞ã)0+∞Zå)dx(0 ≤ α < +∞); ä)(x − a)α + 1arctgxdx (1 < y < +∞);yxy1xcos(x2 + y)dx (y > 0); æ)x+yZ1y cos1dx (|y| < +∞);x201Z1Z1xy−1 ln(1 − x)dx (y > 0); è)ç)Z+∞0sin1 dx·(0 < α < 2).x xα0Z+∞21. Ïóñòüf (x)dx ñõîäèòñÿ (a ≥ 0). Äîêàçàòü, ÷òî íà [0; +∞) ðàâa3. Èíòåãðàëû, çàâèñÿùèå îò ïàðàìåòðà53Z+∞Z+∞2−αxíîìåðíî ñõîäÿòñÿ èíòåãðàëû: à)e f (x)dx; á)e−αx f (x)dx.aaZ+∞22.
Ïóñòüf (x)dx ñõîäèòñÿ àáñîëþòíî. Äîêàçàòü, ÷òî ïðè |y| < +∞aZ+∞ðàâíîìåðíî è àáñîëþòíî ñõîäÿòñÿ èíòåãðàëû: à)f (x) sin(x2 + y 2 )dx;aZ+∞f (x) arctg(xy)dx.á)a23. Ïóñòü f : [a; +∞) → R (a ≥ 0), èíòåãðèðóåìà íà [a; A] äëÿ ëþáîãîZ+∞Z+∞f (x)dx ñõîäèòñÿ. Òîãäàf (xy)dx ñõîäèòñÿ ðàâíîìåðíî íàA(> a) èa[y0 ; +∞) (y0 > 0). Äîêàçàòü.aZ124. Èññëåäîâàòü ôóíêöèþ F (y) =xπ/4 (x2dxíà íåïðåðûâ+ y 2 + 1)0íîñòü.25.
Ïðîâåðèòü íåïðåðûâíîñòü íà óêàçàííûõ ìíîæåñòâàõ ñëåäóþùèõôóíêöèé:Z+∞Z+∞cos(ax)−(x−a)2à) F (a) =edx (a ∈ R); á) F (a) =dx (a ∈ R);1 + x2−∞Z+∞0sin(αx2 )dx (α ∈ (0; +∞));â) F (a) =0Z1ã) F (a) =sin(a/x)dx (a ∈ (0; 1));xa0Z+∞ä) F (a) =2e−(x+a)√√dx (a ∈ [1; 2]);x+ a+10Z1å) F (a) =0ln(ax)√dx (a ∈ [1; +∞));x+aZπæ) F (a) =sin xdx(a ∈ (0; 2)); ç) F (a) =xa (π − x)aZπ00Z+∞è) F (a) =0dx(a ∈ [0; 1));sina xxdx(a ∈ (2; +∞)); ê) F (a) =2 + xaZ+∞0e−x dx(a ∈ (0; 1)).|sin x|a54Îãëàâëåíèå26. Ïóñòü f : [0; +∞) → R èíòåãðèðóåìà íà [0; A] äëÿ ëþáîãî A > 0.Z+∞Òîãäà ôóíêöèÿ F (y) =f (xy)dx íåïðåðûâíà íà (0; +∞).y27. Âû÷èñëèòü:Z+∞Z+∞ −t2 (x2 +1)sin(2tx)e2à) lime−xdx; á) limdx.t→+∞t→+∞xx2 + 10Z+∞28.
Äîêàçàòü, ÷òî: à) limn→+∞Z+∞â) lima→+∞0dx= 1; á) limna→+∞x +10Z+∞cos xdx√ ·= 0; ã) lima→+∞x 1 + a2 x211Z+∞ae−x dx = 1;0arctg(ax)π√dx = .2x2 x2 − 1Z+∞29. Çàêîíåí ëè ïåðåõîä ê ïðåäåëó ïðè α → 0 â èíòåãðàëåαe−αx dx?030. Äîêàçàòü, ÷òî åñëè f íåïðåðûâíà è îãðàíè÷åíà íà [0; +∞), òîZ+∞2af (x)limdx = f (0).a→+0 πx2 + a2031. Äîïóñòèìà ëè ïåðåñòàíîâêà ïîðÿäêà èíòåãðèðîâàíèÿ â èíòåãðàZ+∞ Z+∞ 2Z+∞ Z1y − x2y−xëàõ: à)dydx;á)dydx?(x2 + y 2 )2(x + y)31110Z+∞32. Äîêàçàòü, ÷òî ôóíêöèÿ F (a) =0ôåðåíöèðóåìà íà R.Zπ/233. Âû÷èñëèòü èíòåãðàëû: à)Z1á)0Z1ã)sin xdxíåïðåðûâíà è äèô1 + (x − a)20xa − xbdx (a, b > 0); â)ln xZ1arctg(a tg x)dx;tg xµ1sin lnx¶xa − xbdx (a, b > 0);ln x0µ1cos lnx¶xa − xbdx (a, b > 0); ä)ln x0Z+∞34. Âû÷èñëèòü èíòåãðàë0Zπ/2ln(1 + a cos x)dx (|a| ≤ 1).cos x0arctg(ax)dx.x(1 + x2 )35.
Èñïîëüçóÿ ðåçóëüòàò ïðèìåðà 34, ïîêàçàòü, ÷òî:Zπ/2Zπ/2ππtg xdx = ln 2; á)ln sin xdx = − ln 2.à)x22003. Èíòåãðàëû, çàâèñÿùèå îò ïàðàìåòðàZ+∞36. Âû÷èñëèòü èíòåãðàëû: à)5522e−ax − e−bxdx (a, b > 0);x20Z+∞Z+∞22á)e−(ax +bx+c) dx (a > 0); â)e−ax ch(bx)dx (a > 0);−∞Z+∞2e−axã)0Z+∞sh(bx)dx (a > 0); ä)x2e−ax cos(bx)dx (a > 0);00Z+∞Z+∞¡ −α/x22¢−ax2 sin(bx)eå)dx (a > 0); æ)e− e−β/x dx (α, β > 0).x00Z+∞37. Âû÷èñëèòü èíòåãðàëû: à)Z+∞á)022cos (ax) − cos (bx)dx; â)x0Z+∞ã)2 sin(ax) − sin(2ax)dx; ä)x30Z+∞å)ç)Z+∞cos(ax) + cos(bx) − 2dx;x20Z+∞sin(ax) sin(bx)dx;x20Z+∞sin4 (ax) − sin4 (bx)dx; æ)x0Z+∞cos(ax) − cos(bx)dx;x2sin(x3 )dx; è)x0Z+∞ë)0Z+∞0sin5 xdx; ê)x0sin x − x cos xdx; ì)x3Z+∞sin4 (ax)dx;x4Z+∞x − sin xdx;x30sin(ax) −bxe dx (b > 0);x0Z+∞Z+∞ −axdxe− e−bxí)e−ax sin2 (bx)(a > 0); î)cos xdx (a, b > 0);xx0Z+∞ï)01 − cos(ax) −bxe dx (b > 0).x0Z+∞38.
Âû÷èñëèòü èíòåãðàëû: à)Z+∞á)sin2 (ax)dx;1 + x20cos(px)dx (a > 0, ac − b2 > 0); â)2ax + bx + c−∞Z+∞0sin2 (ax)dx.x2 (1 + x2 )Z+∞Z+∞dxq39. Âû÷èñëèòü èíòåãðàëû: à)xp e−x dx; á)(ln x)p 2 ;x0156ÎãëàâëåíèåµZ1â)x31lnx¶5−∞0Z2å)Z+∞Z2dx−ex ppdx; ã)e e xdx; ä);32x (2 − x)dxp4−1Zπè)(2 − x)(1 + x)3sinp xdx; ê)1 + cos x0Z+∞ë)Z1; æ)0Z1 µln1x0xdxp; ç)(2 − x) 3 x2 (1 − x)¶pµln ln1x¶Z+∞xp−1 dx;1 + xq0dx;0ln xdx; ì)x2 + a2Z+∞ln2 xdx.1 + x400ZZ40. Äîêàçàòü ñõîäèìîñòü èíòåãðàëàK = {(x, y) : 0 ≤ x ≤ 1; 0 ≤ y ≤ 1}.cos(xy)dxdy , ãäåx+yKZZ41.
Èññëåäîâàòü íà ñõîäèìîñòü èíòåãðàëdxdy, ãäå:|x|p + |y|pEà) E = {(x, y) : |x| + |y| ≥ 1}; á) E = {(x, y) :Z Z|x| + |y| ≤ 1} (p > 0).dxdy42. Èññëåäîâàòü íà ñõîäèìîñòü èíòåãðàë, ãäå:|x|p + |y|qEà) E = {(x, y) : |x| + |y| ≥ 1}; á) ZEZ= {(x, y) : |x| + |y| ≤ 1} (p, q > 0).43.
Óáåäèòüñÿ, ÷òî èíòåãðàësin(x2 + y 2 )dxdy ðàñõîäèòñÿ, ïðîâåðèâ, ÷òî:à) limR2ZZn→+∞|x|≤nsin(x2 + y 2 )dxdy = π ;ZZá) limn→+∞x2 +y 2 ≤2πnsin(x2 + y 2 )dxdy = 0.ZZ44. Ïîêàçàòü, ÷òîx2 − y 2dxdy , ãäå K = {(x, y) : x ≥ 1, y ≥ 1},(x2 + y 2 )2KZ+∞ Z+∞ 2x − y2dydx,ðàñõîäèòñÿ, â òî âðåìÿ êàê ïîâòîðíûå èíòåãðàëû(x2 + y 2 )2Z+∞ Z+∞ 2x − y2dxdy îáà ñõîäÿòñÿ.(x2 + y 2 )21145.ZÂû÷èñëèòüèíòåãðàëû:Zdxdypà), E = {(x, y) : x2 + y 2 < 1};1 − x2 − y 2ZEZdxdyá), E = {(x, y) : y ≥ x2 + 1};x4 + y 2E113.
Èíòåãðàëû, çàâèñÿùèå îò ïàðàìåòðàZZâ)qdxdy22½¾x2 y 2, E = (x, y) : x ≥ 0, y ≥ 0, 2 + 2 < 1 ;ab1 − xa2 − yb2ZZZdxdydzã), K = {(x, y, z); 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1}.xp y q z rEK5758Îãëàâëåíèågamma_f.epsÐèñ. 1: Ãðàôèê Ãàììà-ôóíêöèèËèòåðàòóðà[1] Â.À. Èëüèí, Ý.Ã. Ïîçíÿê, Îñíîâû ìàòåìàòè÷åñêîãî àíàëèçà. ×àñòèI,II, Ì.:Íàóêà, 1971, 1973.[2] Â.À. Èëüèí, Â.À. Ñàäîâíè÷èé, Áë.Õ. Ñåíäîâ, Ìàòåìàòè÷åñêèé àíà-ëèç, Ì.:Íàóêà, 1979.[3] Ã.Ì. Ôèõòåíãîëüö, Êóðñ äèôôåðåíöèàëüíîãî è èíòåãðàëüíîãî èñ÷èñ-ëåíèÿ.
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