ЛЕКЦІЇ ² ВПРАВИ

Ïðèêëàä 8.3.4. Çíàéòè ³íòåãðàë
Ðîçâ’ÿçàííÿ
ψ′
∫ dx . ψ
ψ′
dt
∫ dx = ⎡⎣t =ψ ( x) , dt =ψ ′( x) dx⎤⎦
= ∫ = ln t + C = ln ψ ( x)
+ C .
ψ
t
Îòæå,
ψ′
∫ dx = ln ψ ( x)
+ C .
ψ
Ó â³äïîâ³äíîñò³ äî îñòàííüî¿ ôîðìóëè ìîæíà ñôîðìóëþâàòè
òàêå ïðàâèëî: ÿêùî ï³ä³íòåãðàëüíà ôóíêö³ÿ ÿâëÿº ñîáîþ
äð³á, ó ÿêîìó ÷èñåëüíèê º ïîõ³äíà â³ä çíàìåííèêà, òî
ïåðâ³ñíà â³ä íüîãî äîð³âíþº íàòóðàëüíîìó ëîãàðèôìó ìîäóëÿ
çíàìåííèêà.
Ðîçãëÿíåìî òåïåð êîíêðåòèçîâàí³ ïðàêòè÷í³ ïðèêëàäè.
Ïðèêëàä 8.3.5. Çíàéòè ³íòåãðàëè: 1)
Ðîçâ’ÿçàííÿ
1)
′
cos x ( sin x)
∫ctgxdx = ∫ dx = ∫ dx = ln sin x + C;
sin x sin x
∫ ñtg xdx ; 2) ∫ tg xdx .
(8.3.5)
′
sin x ( cos x)
2) ∫tg xdx = ∫ dx =− ∫ dx =− ln cos x + C.
(8.3.6)
cos x cos x
Çàóâàæåííÿ. Îäåðæàí³ ôîðìóëè (8.3.5) – (8.3.6) ìîæíà
çàíåñòè ó òàáëèöþ íåâèçíà÷åíèõ ³íòåãðàë³â.
2. Äðóãèé òèï ï³äñòàíîâêè. Íåõàé çàäàíî ³íòåãðàë
∫ f( x)
dx.
Òåïåð çðîáèìî çàì³íó: x = ϕ(t), äå ôóíêö³ÿ ϕ(t) ìຠïîõ³äíó
â äåÿêîìó ³íòåðâàë³. Òîä³ çàäàíèé ³íòåãðàë íàáóâຠâèãëÿäó:
( ) = ( ϕ( )) ϕ′ ()
∫f x dx ∫ f t t dt. (8.3.7)
Ñïðàâåäëèâ³ñòü ôîðìóëè (8.3.7) âñòàíîâèìî òàêîæ øëÿõîì
äèôåðåíö³þâàííÿ îáîõ ÷àñòèí ð³âíîñò³ (8.3.7).
Ìàºìî çà îçíà÷åííÿì
′
∫ = . (8.3.8)
( f( x)
dx) f( x)
Ùîäî äèôåðåíö³þâàííÿ ïðàâî¿ ÷àñòèíè ð³âíîñò³ (8.3.7),
òî ìè öþ îïåðàö³þ çä³éñíèìî çà äîïîìîãîþ ôîðìóëè çíàõîäæåííÿ
ïîõ³äíî¿ ñêëàäåíî¿ ôóíêö³¿, ïðè öüîìó ìàòèìåìî
( ∫ ( ()) () ) ∫ ( ()) ()
( ) x ( ()) ()
/ /
f ϕ t ϕ ′ t dt = f ϕ t ϕ′ t dt ⋅ t = f ϕ t ϕ′
t ⋅ =
x
t
ϕ′
( ()) ( )
/ 1
() t
= f ϕ t = f x . (8.3.9)
Ïîð³âíþþ÷è ñï³ââ³äíîøåííÿ (8.3.8) – (8.3.9), âïåâíþºìîñÿ
ó ñïðàâåäëèâîñò³ ôîðìóëè (8.3.7).
Çàóâàæåííÿ. Äðóãèé òèï ï³äñòàíîâêè ïîòðåáóº ï³ñëÿ
³íòåãðóâàííÿ ïðàâî¿ ÷àñòèíè ïîâåðíåííÿ äî ñòàðî¿ çì³ííî¿.
ßê â³äîìî, öå áóäå ãàðàíòîâàíî, ÿêùî áóäå ³ñíóâàòè îáåðíåíà
ôóíêö³ÿ äëÿ ôóíêö³¿ x = ϕ(t).
2 2
Ïðèêëàä 8.3.6. Çíàéòè ³íòåãðàë ∫ 7 − x dx .
Ðîçâ’ÿçàííÿ
∫
⎡ x= 7sin t, dx=
7cos tdt;
⎤
2 2
7 − xdx= ⎢
⎥ =
2 2
⎢⎣
7 − x = 7 cost = 7 cos t,cos t ≥0⎥⎦
2 49 49 1
x
= 49∫cos tdt = ∫( 1+ cos 2 t)
dt = ( t + sin 2 t) + C, t = arcsin .
2 2 2 7
ÂÏÐÀÂÈ
Çíàéòè ³íòåãðàëè:
8.8.
dx
∫ ; 8.9.
x
3 5
∫
dt
3−
4t
2
−
; 8.10. ∫ cos3ϕd
ϕ; 8.11. 2
∫ e dx;
x
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