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Ïðîèíòåãðèðóåì c ïîìîùüþ çàìåíûy = cos x.Ñëó÷àé III. Ïóñòü ôóíêöèÿ f π-ïåðèîäè÷íà. Òîãäà âñå îäíî÷ëåíû ìíîãî÷ëåíîâP è Q ÷åòíûõ ñòåïåíåé (èëè âñå íå÷åòíûõ, íî òîãäà äîìíîæèì ÷èñëèòåëü èçíàìåíàòåëü íà cos x).Åñëè çíàìåíàòåëÿ íåò (Q ≡ 1), ïðèìåíÿþò ôîðìóëû ïîíèæåíèÿ ñòåïåíè:cos2 x =1+cos 2x1−cos 2xsin 2x, sin x cos x =., sin2 x =222Âîçìîæíî,ïîíèæåíèå ñòåïåíè ïðèäåòñÿ ïðèìåíèòü íå îäèí ðàç (íàïðèìåð, äëÿRcos4 x dx), íî ìû îáÿçàòåëüíî ïðèäåì ê òîìó, ÷òî â ëþáîì îäíî÷ëåíå áóäåòâñòðå÷àòüñÿ ëèáî êîñèíóñ, ëèáî ñèíóñ â íå÷åòíîé ñòåïåíè (ñëó÷àè I è II).Åñëè æå çíàìåíàòåëü íåòðèâèàëåí, ñëåäóåò ñäåëàòü çàìåíó t = tg x èëè t =ctg x.10Ïðèìåð 3.2.Zdx=sin4 xZ1dx=sin2 x tg2 x cos2 xZcos−2 xd tg x =tg4 x1 + t2t−31dt=−− t−1 + Cn = − ctg3 x − ctg x + Cn .4t33Z=Êîíñòàíòû Cn íåçàâèñèìû, êàæäàÿ ñîîòâåòñòâóåò ñâîåìó èíòåðâàëó (πn; π(n+1)).Îáùèé ñëó÷àé. Åñëè ôóíêöèÿ f (x) íå îáëàäàåò âûøåóïîìÿíóòûìè ñèììåòðèÿìè:f (π − x) ≡ −f (x), f (−x) ≡ −f (x) èëè f (π + x) ≡ f (x),òî èìååòñÿ åùå îäèí ñïîñîá ñâåñòè èíòåãðèðîâàíèå òðèãîíîìåòðè÷åñêîé ôóíêöèè ê èíòåãðèðîâàíèþ ðàöèîíàëüíîé óíèâåðñàëüíàÿ òðèãîíîìåòðè÷åñêàÿ ïîäñòàíîâêà.
Ýòîò ìåòîä ïîäõîäèò äëÿ âñåõ ñëó÷àåâ, íî áîëååx òðóäîåìîê.Íà êàæäîì èíòåðâàëå (2πn − π; 2πn + π) ïîëîæèì t = tg 2 . Òîãäàx = 2πn + 2 arctg t =⇒ dx =sin x =Ïðèìåð 3.3.Zdx=2 + sin xZ2dt;1 + t22t1 − t22t; cos x =; tg x =.21+t1 + t21 − t22dt/(1 + t2 )=2t2+1Z + t2Zdtdt===1 + t2 + t(t + 1/2)2 + 3/4x2 tg + 12(t + 1/2)22√√2+ Cn = √ arctg+ Cn .= √ arctg3333Êîíñòàíòû Cn ñîîòâåòñòâóþò ðàçíûì èíòåðâàëàì (2πn− −π; 2πn + π). Îäíàêîïîäûíòåãðàëüíàÿ ôóíêöèÿ íåïðåðûâíà â òî÷êàõ 2πn+π, ïîýòîìó åå ïåðâîîáðàçíàÿF (x) äîëæíà áûòü íåïðåðûâíà â íèõ.
Çíà÷èò, êîíñòàíòû Cn æåñòêî ñâÿçàíûìåæäó ñîáîé. Âû÷èñëèì F (2πn + π), èñõîäÿ èç íåïðåðûâíîñòè F ñëåâà è ñïðàâà:limx→2πn+π−0limπ2F (x) = √ arctg(+∞) + Cn = √ + Cn ,33−π2F (x) = √ arctg(−∞) + Cn+1 = √ + Cn+1 .332πn2π= Cn + √ =⇒ Cn = C0 + √33x→2πn+π+0.Ñëåäîâàòåëüíî, Cn+1Òåïåðü ðàññìîòðèì äîïîëíèòåëüíûé ïðèåì èíòåãðèðîâàíèÿ òðèãîíîìåòðè÷åñêèõ ôóíêöèé ïðåîáðàçîâàíèå ïðîèçâåäåíèÿ â ñóììó. Îí îñíîâàí íà òðåõ òîæäåñòâàõ:cos x cos y =cos(x − y) + cos(x + y);211cos(x − y) − cos(x + y);2sin(x + y) + sin(x − y)sin x cos y =.2sin x sin y =Ïðèìåð 3.4.Zsin x cos 3x sin 4x dx ==1412ZZ(sin 4x − sin 2x) sin 4x dx =1 − cos 8x − cos 2x + cos 6x dx =sin 8x sin 2x sin 6x 1x−−++ C.=4826Çàäà÷è ïî òåìå ëåêöèè 3Ïðîèíòåãðèðîâàòü ôóíêöèþ, èñïîëüçóÿ çàìåíó y = sin x èëè y = cos x:Z3.1.sin5 x dx.
3.2.Ztg x sin2 x dx. 3.3.Zcos xdx.4 + sin2 xÏðîèíòåãðèðîâàòü ôóíêöèþ, èñïîëüçóÿ ôîðìóëû ïîíèæåíèÿ ñòåïåíè:Z3.4.cos4 x dx. 3.5.Zsin2 x cos2 x dx.Ïðîèíòåãðèðîâàòü ôóíêöèþ, èñïîëüçóÿ çàìåíó t = tg x èëè t = ctg x:Z3.6.Z3tg x dx. 3.7.dx.1 + 3 cos2 xÏðîèíòåãðèðîâàòü ôóíêöèþ, èñïîëüçóÿ çàìåíó t = tg x2 :Z3.8.dx. 3.9.2 sin x + cos xZdx.3 + cos xÏðîèíòåãðèðîâàòü ôóíêöèþ, ïðåîáðàçóÿ ïðîèçâåäåíèå â ñóììó:Z3.10.sin 2x cos 3x dx.ËåêöèÿN o 4.Èíòåãðèðîâàíèå èððàöèîíàëüíûõôóíêöèéËèíåéíûå èððàöèîíàëüíîñòèÔóíêöèè âèäàãäå√k√kx+b√,f (x) =Q x, k x + bP x,è Q(u, v) ìíîãî÷ëåíû, èíòåãðèðóþòñÿ ïðè ïîìîùè çàìåíû.
Òîãäà x = tk − b, dx = ktk−1 dt, è èíòåãðàë ïðèìåò âèäP (u, v)x+bZZf (x)dx =t =ZP tk − b, t k−1P1 (t)ktdt=dt,Q (tk − b, t)Q1 (t)ãäå P1 (t) è Q1 (t) ìíîãî÷ëåíû. Ïîëó÷èëñÿ èíòåãðàë îò ðàöèîíàëüíîé ôóíêöèè.Ïðèìåð 4.1.√ . Ïîñêîëüêó ïðèñóòñòâóþò êîðíè 2-é è 3-éÂû÷èñëèòü èíòåãðàë √x dx+ x√ñòåïåíåé, çäåñü íàäî âçÿòü t = x, òîãäà x = t6 , dx = 6t5 dt.Z36Z√ZZ 3dx6t5 dtt dt√=6==t 3 + t2t+1x+ 3xZ1= 6 t2 − t + 1 −dt = 2t3 − 3t2 + 6t − 6 ln |t + 1| + C =t+1√√√√= 2 x − 3 3 x + 6 6 x − 6 ln( 6 x + 1) + C.Êâàäðàòè÷íûå èððàöèîíàëüíîñòèÐàññìîòðèì ôóíêöèè âèäà pP x, ±(x2 + px + q),f (x) = pQ x, ±(x2 + px + q)(3)ãäå P (u, v) è Q(u, v) ìíîãî÷ëåíû. Íà÷íåì ñ íàèáîëåå ïðîñòûõ ñëó÷àåâ.Ñëó÷àé I.
Ïóñòü f (x) = p±(xP2 +(x)px + q) , ãäå P (x) ìíîãî÷ëåí. Èíòåãðàë ìîæíîíàéòè â âèäåZZpf (x)dx = R(x) ±(x2 + px + q) + pk dx±(x2 + px + q)13,(4)ãäå R(x) ìíîãî÷ëåí ìåíüøåé ñòåïåíè, ÷åì P (x). Åãî íàõîäÿò ìåòîäîì íåîïðåäåëåííûõ êîýôôèöèåíòîâ, ïðîäèôôåðåíöèðîâàâ îáå ÷àñòè (4) è âû÷èñëÿÿ êîýôôèöèåíòûR(x) ïîñëåäîâàòåëüíî îò ñòàðøåãî ê ìëàäøèì.Ïðèìåð 4.2.Z px2 + 1 dx =Zx2 + 1√dx =x2 + 1Zpdx2.= (Ax + B) x + 1 + k √x2 + 1Çäåñü deg P =√2 =⇒ deg R 6 1. Ïðîäèôôåðåíöèðîâàâ ëåâóþ è ïðàâóþ ÷àñòè èäîìíîæèâ íà x2 + 1, ïîëó÷àåì1x2 + 1 = A(x2 + 1) + (Ax + B)2x + k = 2Ax2 + Bx + A + k,2ñëåäîâàòåëüíî,= 0, k = 1/2.Z p A = 1/2, B ppxÎòâåò: x2 + 1 dx = 2 x2 + 1 + 12 lnx + x2 + 1 + C .Ñëó÷àé II.
Ïóñòü f (x) = x2=⇒ dx = −dt/tZxm√è ïîëó÷èìdx=ax2 + bx + cZm√√,1m∈N+ bx + cax2. Ñäåëàåì ïîäñòàíîâêó x = 1/t−tm−2 dt=at−2 + bt−1 + cZ=−sign t · tm−1 dt√.a + bt + ct2Òàêèì îáðàçîì, ïðè c 6= 0 ìû ïðèøëè ê ñëó÷àþ I, à ïðè c = 0 ê ëèíåéíîéèððàöèîíàëüíîñòè.III. Îáùèé ñëó÷àé. Ëèíåéíîé çàìåíîé y = x + p/2 âûðàæåíèå (3) ñâîäèòñÿ êâèäó 2pa − y 2 (IIIa)P y, Z(y)pg(y) = , ãäå Z(y) = y 2 + a2 (IIIá)Q y, Z(y)y 2 − a2 (IIIâ)Ðàññìîòðèì ýòè òðè ïîäñëó÷àÿ.(IIIa). Îáëàñòü îïðåäåëåíèÿ pa2 − y2 îòðåçîê [−a; a]p.Ïîäñòàíîâêà: y = a sin t, |t| 6 π/2. Òîãäà dy = a cos t dt, a2 − y2 = a cos t.
Ïîëó÷àåìpZZP y,Z(y)p dy =Q y, Z(y)P (a sin t, a cos t) a cos tdt,Q (a sin t, a cos t)òàêèå èíòåãðàëû áûëè èçó÷åíû íà ëåêöèè 3. Íàêîíåö, âûðàæàåì t ÷åðåç y: t =arcsin(y/a).14Ïðèìåð 4.3.ïóñòüZ p3R2 − x2 dx =x = R sin t, |t| 6 π/2ZZR4= (R cos t)3 R cos t dt =(1 + cos 2t)2 dt =4Z1 + cos 4tR4 sin 4t R4=1 + 2 cos 2t +dt =3t + 2 sin 2t +.4284xt = arcsinR√ x 222R2 − 2x22x R − xsin 2t =, cos 2t = 1 − 2=2RRR2√4x R2 − x2 (R2 − 2x2 ).sin 4t =R4Îñòàëîñü ïîäñòàâèòü â ýòî âûðàæåíèå,è(IIIá). Îáëàñòü îïðåäåëåíèÿ py âñÿ îñü R. Ìîæíî ñâåñòè ê èíòåãðàëóîò òðèãîíîìåòðè÷åñêîé ôóíêöèè ïîäñòàíîâêîé2+ a2y = a tg t =⇒ dy =a dt;cos2 tpy 2 + a2 =acos tèëè ñðàçó ïðèéòè ê èíòåãðàëóðàöèîíàëüíîé ôóíêöèè, ñäåëàâ ïåðâóþ ïîäñòàíîâêóa1Ýéëåðà (ðèñ.
À): y = 2 t − t , t > 0. Òîãäà1a1 + 2 dt,dy =2tpa1y 2 + a2 =t+,2tt=y+py 2 + a2.aÏðèìåð 4.4.Çäåñü ñäåëàåì ïîäñòàíîâêó x = 2 tg t; dx = 2 dt/ cos2 t.Zdx=(4 + x2 )3/2ZZcos3 t 2 dtcos tsin t=dt =+C =28 cos t441xxxx+ C.= sin arctg + C = cos arctg + C = √42824 4 + x215Ïðèìåð 4.5.Çäåñü ñëåäóåò ïðèìåíèòü ïåðâóþ ïîäñòàíîâêó Ýéëåðà x = 12Zdx√=1 + 1 + x2(t2 + 1)dt=t(t + 1)2Z=Zt−,.1t>0t1(1 + t−2 )dt2=11 + (t + t−1 )2Z122−dt=lnt++C√t(t + 1)2t+1t=x+ 1+x2 .py 2 − a2(−∞; −a] [a; +∞)(IIIâ). Îáëàñòü îïðåäåëåíèÿ äâà ëó÷àè. Ìîæíîñâåñòè ê èíòåãðàëó îò òðèãîíîìåòðè÷åñêîé ôóíêöèè, ñäåëàâ ïîäñòàíîâêóy=aa sin t dt p 2=⇒ dy =;y − a2 = a tg t.cos tcos2 tÊðîìå òîãî, ìîæíî ñðàçó ñâåñòè ê ðàöèîíàëüíîé ôóíêöèè, ñäåëàâ âòîðóþ ïîäñòàíîâêó Ýéëåðà (ðèñ.
Á):y=1a1 pa at+, |t| > 1; dy =1 − 2 dt, y 2 − a2 = t −2t2t2Íî çäåñü âûðàæåíèå t ÷åðåç y ìåíåå óäîáíî:1 .tpy + sign y y 2 − a2.t=aÏðèìåð 4.6.Ïóñòü x > 0. Cäåëàåì ïîäñòàíîâêó x = 1/ cos t, ãäå 0 6 6 t < π/2; ñîîòâåòñòâåííî dx = sin t dt/cos2 t:Z √x2 − 1dx =xÔóíêöèè âèäàZZtg t sin t dt=tg2 t dt =cos−1 t cos2 tZp11=−1dt=tgt−t+C=x2 − 1 − arccos + C.cos2 txÄðîáíî-ëèíåéíûå èððàöèîíàëüíîñòèrax + bP x,αx + βf (x) = r,ax + bQ x, kαx + β a b 6= 0∆=α β kãäå P (u, v) è Q(u, v) ìíîãî÷ëåíû,çàìåíû t =skax + bαx + β.16, èíòåãðèðóþòñÿ c ïîìîùüþÂûðàçèì x ÷åðåç t:tk =ax + bβtk − b=⇒ αtk x + βtk − ax − b = 0 =⇒ x =;αx + βa − αtkdx =k∆ tk−1 dt.(αtk − a)2Ïîëó÷àåì èíòåãðàë ðàöèîíàëüíîé ôóíêöèè:βtk − bZZP, t tk−1a − αtkdt.f (x)dx = k∆ kβt − bk2, t (αt − a)Qa − αtkÂïðî÷åì, ïðè k = 2 ìîæíî ñâåñòè äðîáíî-ëèíåéíóþ èððàöèîíàëüíîñòü ê êâàäðàòè÷íîé, íàïðèìåð,r|x + 2|x+2x+2.=p= sign(x + 2) px−1(x + 2)(x − 1)(x + 2)(x − 1)Èíòåãðàëüíûé áèíîìÅñëè ìû õîòèì ïðîèíòåãðèðîâàòü ôóíêöèþ âèäàf (x) = xp√nkxq + a ,ãäå k, n ∈ N, p, q ∈√Q, òî âîçìîæíû òðè ñëó÷àÿ:1) çàìåíà y = √xq + a ñâåäåò ê èíòåãðàëó îò ðàöèîíàëüíîé ôóíêöèè;2) çàìåíà y = 1 + ax−q ñâåäåò ê èíòåãðàëó îò ðàöèîíàëüíîé ôóíêöèè;3) åñëè íè òà, íè äðóãàÿ çàìåíà íå ïðèâîäèò ê öåëè, òî èíòåãðàë íåáåðóùèéñÿ.Êàê ïðèìåð äëÿ ñëó÷àÿ 1) ïðåäëàãàåòñÿ çàäà÷à 4.9.nnÇàäà÷è ïî òåìå ëåêöèè 4Ïðîèíòåãðèðîâàòü ëèíåéíûå èððàöèîíàëüíîñòè:Z4.1.√xdx.
4.2.x+1Zdx√ dx.x+ xÏðîèíòåãðèðîâàòü êâàäðàòè÷íûå èððàöèîíàëüíîñòè ìåòîäîì íåîïðåäåëåííûõêîýôôèöèåíòîâ:Z4.3.√17 − 2x2dx. 4.4.5 + 4x − x217Z2x2 − x√dx.x2 − 4Ïðîèíòåãðèðîâàòü êâàäðàòè÷íûå èððàöèîíàëüíîñòè ñ ïîìîùüþ ïîäõîäÿùèõçàìåí ïåðåìåííûõ:Z4.5.Z4.7.2xpx2Z− 1 dx. 4.6.dx√dx. 4.8.1 + 2x − x2Âû÷èñëèòü èíòåãðàëüíûå áèíîìû:Z4.9.Z(x2 + 1)3/2 dx.4dx√dx.x + x2 + 4x(ðåøåíèå ñì. â ïðèìåðå 10.1).1p1 + x−4 dxxZp34.10.
x5 x3 + 1 dx.ËåêöèÿN o 5.Èíòåãðàë ÐèìàíàÏóñòü ôóíêöèÿ f îïðåäåëåíà íà îòðåçêå [a; b].Îïðåäåëåíèå5.1. Ðàçáèåíèåì îòðåçêà [a; b] íàçûâàåòñÿ íàáîð òàêèõ òî÷åê{ai }ni=0 , n ∈ N, ÷òîa = ao < a1 < a2 < . . . < an = b.Äèàìåòð ðàçáèåíèÿ diam{a } = max(a − a ).Îïðåäåëåíèå 5.2. Ðàçáèåíèå {b } íàçûâàåòñÿ èçìåëü÷åíèåì ðàçáèåíèÿ {a } ,niii=1i−1mj j=0åñëè {ai }ni=0 ⊂ {bj }mj=0 , ò.å.ni i=0∀i ∈ {0, .
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