Integruvannya chastinami odin iz sposobiv znahodzhennya integrala Sut metodu v nastupnomu yaksho pidintegralna funkciya podana u vidi dobutku dvoh neperervnih i gla dkih funkcij kozhna z yakih mozhe buti yak elementarnoyu funkciyeyu tak i kompoziciyeyu to spravedlivi formuli dlya neviznachenogo integrala u d v u v v d u displaystyle int u dv u v int v du dlya viznachenogo a b u d v u v a b a b v d u displaystyle int limits a b u dv u v bigg a b int limits a b v du Peredbachayetsya sho znahodzhennya integrala v d u displaystyle int v du prostishe nizh u d v displaystyle int u dv U inshomu vipadku zastosuvannya metodu ne vipravdano Zmist 1 Oderzhannya formul 1 1 Dlya neviznachenogo integrala 1 2 Dlya viznachenogo 2 Prikladi 3 Div takozh 4 DzherelaOderzhannya formul RedaguvatiDlya neviznachenogo integrala Redaguvati Funkciyi u displaystyle textstyle mathit u nbsp i v displaystyle textstyle mathit v nbsp gladki otzhe mozhlive diferenciyuvannya d u v d u v u d v displaystyle d u v du v u dv nbsp Ci funkciyi takozh neperervni otzhe mozhna vzyati integral vid oboh chastin rivnosti d u v d u v u d v displaystyle int d u v int du v int u dv nbsp Operaciya integruvannya protilezhna diferenciyuvannyu u v d u v u d v displaystyle u v int du v int u dv nbsp Pislya perestanovok u d v u v v d u displaystyle int u dv u v int v du nbsp Dlya viznachenogo Redaguvati U cilomu analogichno vipadku dlya neviznachenogo integrala d u v d u v u d v displaystyle d u v du v u dv nbsp a b d u v a b d u v a b u d v displaystyle int limits a b d u v int limits a b du v int limits a b u dv nbsp a b u d v u v a b a b v d u displaystyle int limits a b u dv u v bigg a b int limits a b v du nbsp Prikladi Redaguvati x cos x d x x d sin x x sin x sin x d x x sin x cos x C displaystyle int x cos x dx int x d sin x x sin x int sin x dx x sin x cos x C nbsp e x x d x x e x d x x d e x x e x e x d x x e x e x C displaystyle int e x x dx int x e x dx int x de x x e x int e x dx x e x e x C nbsp Inodi cej metod zastosovuyetsya kilka raziv x 2 sin x d x x 2 d cos x x 2 cos x 2 x cos x d x displaystyle int x 2 sin x dx int x 2 d cos x x 2 cos x int 2x cos x dx nbsp x 2 cos x 2 x d sin x x 2 cos x 2 x sin x 2 sin x d x x 2 cos x 2 x sin x 2 cos x C displaystyle x 2 cos x int 2x d sin x x 2 cos x 2x sin x int 2 sin x dx x 2 cos x 2x sin x 2 cos x C nbsp Cej metod takozh vikoristovuyetsya dlya znahodzhennya integraliv vid elementarnih funkcij ln x d x x ln x 1 x x d x x ln x x C displaystyle int ln x dx x ln x int frac 1 x x dx x ln x x C nbsp arctg x d x x arctg x x 1 x 2 d x x arctg x 1 2 ln 1 x 2 C displaystyle int operatorname arctg x dx x operatorname arctg x int frac x 1 x 2 dx x operatorname arctg x frac 1 2 ln 1 x 2 C nbsp dd U deyakih vipadkah integruvannya chastinami ne daye pryamoyi vidpovidi I 1 e a x sin b x d x displaystyle I 1 int e alpha x sin beta x dx nbsp e a x d 1 b cos b x 1 b e a x cos b x a b e a x cos b x d x 1 b e a x cos b x a b I 2 displaystyle int e alpha x d Big frac 1 beta cos beta x Big frac 1 beta e alpha x cos beta x frac alpha beta int e alpha x cos beta x dx frac 1 beta e alpha x cos beta x frac alpha beta I 2 nbsp I 2 e a x cos b x d x displaystyle I 2 int e alpha x cos beta x dx nbsp e a x d 1 b sin b x 1 b e a x sin b x a b e a x sin b x d x 1 b e a x sin b x a b I 1 displaystyle int e alpha x d Big frac 1 beta sin beta x Big frac 1 beta e alpha x sin beta x frac alpha beta int e alpha x sin beta x dx frac 1 beta e alpha x sin beta x frac alpha beta I 1 nbsp U takij sposib odin integral virazhayetsya cherez inshij I 1 1 b e a x cos b x a b I 2 I 2 1 b e a x sin b x a b I 1 displaystyle begin cases I 1 frac 1 beta e alpha x cos beta x frac alpha beta I 2 I 2 frac 1 beta e alpha x sin beta x frac alpha beta I 1 end cases nbsp Virishivshi otrimanu sistemu oderzhuyemo I 1 e a x a 2 b 2 a sin b x b cos b x C displaystyle I 1 frac e alpha x alpha 2 beta 2 Big alpha sin beta x beta cos beta x Big C nbsp I 2 e a x a 2 b 2 a cos b x b sin b x C displaystyle I 2 frac e alpha x alpha 2 beta 2 Big alpha cos beta x beta sin beta x Big C nbsp Div takozh RedaguvatiMetodi integruvannyaDzherela RedaguvatiGrigorij Mihajlovich Fihtengolc Kurs diferencialnogo ta integralnogo chislennya 2023 1300 s ukr Metodi pidstanovki ta integruvannya chastinami u viznachenomu integrali Visha matematika v prikladah i zadachah Klepko V Yu Golec V L 2 ge vidannya K Centr uchbovoyi literaturi 2009 S 418 594 s Otrimano z https uk wikipedia org w index php title Integruvannya chastinami amp oldid 40287942
