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1.
TÝch ph©n
Ph¬ng ph¸p tÝnh TÝch ph©n Mời Thầy cô vào http://violet.vn/n2chanoi để có nhiều tư liệu cùng loại I. TÝnh tÝch ph©n b»ng ph¬ng ph¸p ®æi biÕn: Nh÷ng phÐp ®æi biÕn phæ th«ng: - NÕu hµm cã chøa dÊu ngoÆc kÌm theo luü thõa th× ®Æt t lµ phÇn bªn trong dÊu ngoÆc nµo cã luü thõa cao nhÊt. - NÕu hµm chøa mÉu sè th× ®Æt t lµ mÉu sè. - NÕu hµm sè chøa c¨n thøc th× ®Æt t lµ phÇn bªn trong dÊu c¨n thøc. dx - NÕu tÝch ph©n chøa th× ®Æt t = ln x . x - NÕu tÝch ph©n chøa e x th× ®Æt t = e x . dx - NÕu tÝch ph©n chøa th× ®Æt t = x . x dx 1 - NÕu tÝch ph©n chøa 2 th× ®Æt t = . x x - NÕu tÝch ph©n chøa cos xdx th× ®Æt t = sin x . - NÕu tÝch ph©n chøa sin xdx th× ®Æt t = cos x . dx - NÕu tÝch ph©n chøa th× ®Æt t = tgx . cos 2 x dx - NÕu tÝch ph©n chøa th× ®Æt t = cot gx . sin 2 x Bµi tËp minh ho¹: dx e x dx 1. ∫ ( x + 1) ( x + 2x − 1) dx 1 1 e 1 3 2. ∫ x. 1 − xdx ∫ x. 4. ∫ x 2 3. 3 0 0 1 1 − ln 2 x 0 e −1 π π π 1 dx ∫ cos xdx 4sin xdx e tgx dx 2 2 3 4 5. 6. ∫ sin 2 x − 5sin x + 6 7. ∫ 1 + cos x 8. ∫ cos 2 x 0 x 1+ x 0 0 0 π dx 2 1 9. ∫ 10. ∫ x . 1 − x dx 3 2 4 π sin x 0 4 II. TÝnh tÝch ph©n b»ng ph¬ng ph¸p tÝch ph©n tõng phÇn: b b b C«ng thøc: ∫ f ( x )dx = uv a − ∫ vdu . Nh vËy viÖc chän ®îc u vµ dv cã a a vai trß quyÕt ®Þnh trong viÖc ¸p dông ph¬ng ph¸p nµy. Ta thng gĆ p ba lo i tÝch ph©n nh ę ¹ sau: Lo¹i 1: -N2C- 1
2.
TÝch ph©n b ∫
Pn ( x). sin f ( x).dx ba ∫ Pn ( x). cos f ( x).dx ⇒ u = Pn ( x ) : Trong ®ã Pn ( x ) lµ ®a thøc bËc n. a b ∫ Pn ( x).e .dx f (x) a Ta ph¶i tÝnh n lÇn tÝch ph©n tõng phÇn. b Lo¹i 2: ∫ P( x). ln n f ( x ).dx ⇒ u = ln n f ( x ) : TÝnh n lÇn tÝch ph©n tõng a phÇn. b αx ∫ e . sin β x.dx Lo¹i 3: b a §©y lµ hai tÝch ph©n mµ tÝnh tÝch ph©n e αx . cos β x.dx ∫ a nµy ph¶i tÝnh lu«n c¶ tÝch ph©n cßn l¹i. Th«ng thêng ta lµm nh sau: b - TÝnh ∫ e . sin β x.dx :§Æt u = e αx . Sau khi tÝch ph©n tõng phÇn ta αx a l¹i cã tÝch ph©n b ∫e αx . cos β x.dx .Ta l¹i ¸p dông TPTP víi u nh trªn. a - Tõ hai lÇn TPTP ta cã mèi quan hÖ gi÷a hai tÝch ph©n vµ dÔ dµng t×m ®îc kÕt qu¶. Bµi tËp minh ho¹: π e π 1. ∫ ( x − x + 1) . sin x.dx 2 ∫x ∫x 3 2 2 2 2. . ln x.dx 3. . cos 3x.dx 1 0 0 π π π 2 2 2 4. ∫ e 3x . cos 5x.dx 5. ∫ e 2003x . sin 2004x.dx 6. ∫ e 2x . sin 2 x.dx 0 0 0 Ngoµi ra ta xÐt thªm mét vµi bµi tÝch ph©n ¸p dông ph¬ng ph¸p TPTP nhng kh«ng theo quy t¾c ®Æt ë trªn: 3 x 8 .dx π e 2 e ln x 1 x 2e x .dx 1. ∫ cos( ln x ) .dx 2. ∫ 4 3. ∫ .dx 4. ∫ 5. 0 ( x − 1) 0 ( x + 2) 3 2 1 1 x π 2 1 + sin x ∫ 1 + cos x .e dx x 0 III. TÝch ph©n hµm ph©n thøc h÷u tû: PhÇn 1: TÝch ph©n h÷u tû c¬ b¶n. A A 1. a.D¹ng: ∫ ax + b dx = a ln ax + b + C -N2C- 2
3.
TÝch ph©n
ax + b a A b.D¹ng: ∫ dx = ∫ dx + ∫ dx cx + d c cx + d ax 2 + bx + c C c. D¹ng: ∫ dx = ∫ ( Ax + B ) dx + ∫ dx dx + e dx + e dx 2. a.D¹ng: ∫ 2 ax + bx + c dx 1 ( x − x1 ) − ( x − x 2 ) dx - NÕu ∆ > 0 : ∫ = ∫ a( x − x )( x − x ) = ... a( x − x 1 )( x − x 2 ) x 2 − x 1 1 2 dx ∫ 2 = ... - NÕu ∆ = 0 : b a x − 2a dx - NÕu ∆ < 0 : ∫ ( x − α ) = β.tgt ( x − α ) 2 + β 2 §Æt Ax + B 3. D¹ng: I = ∫ 2 dx ax + bx + c Ph©n tÝch: I = ∫ 2 Ax + B dx = m .∫ ( ax 2 + bx + c)' dx + n. dx ax + bx + c ax + bx + c 2 ∫ ax 2 + bx + c dx = m . ln ax 2 + bx + c + n.∫ 2 ax + bx + c Bµi tËp minh ho¹: 1 2004x − 2003 2 dx 4 dx 1 dx 1. ∫ dx 2. ∫ 3. ∫ x 2 − 6x + 9 4. ∫x 0 2003x + 2004 1 6 + x + 5x + x+1 2 2 0 0 2 2x + 3 1 4 − 3x 5. ∫ dx 6. ∫ 2 dx 1 6 + x + 5x 0 x + x + 1 2 b A( x ) PhÇn 2: TÝch ph©n h÷u tû tæng qu¸t. ∫ Q(x) dx a - Bíc 1: NÕu bËc cña A(x) lín h¬n bËc cña B(x): Chia chia A(x) cho B(x). Ta ph¶i tÝnh tÝch ph©n: b P( x ) ∫ Q(x) dx a - Bíc 2: + NÕu Q(x) chØ toµn nghiÖm ®¬n: Q( x ) = ( x − a 1 )( x − a 2 ) ...( x − a n ) , ta t×m A 1 ,A 2 ...A n sao cho : P( x ) A1 A2 An = + + .. + Q( x ) x − a 1 x − a 2 x − an + NÕu Q(x) gåm c¶ nghiÖm ®¬n vµ nghiÖm béi: Q( x ) = ( x − a )( x − b )( x − c ) , ta t×m A, B,C1 ,C 2 sao cho : 2 P( x ) A B C1 C2 = + + + Q( x ) x − a x − b ( x − c ) 2 ( x − c) + NÕu Q(x) gåm nh©n tö bËc hai ®¬n vµ nh©n tö bËc hai ®¬n: -N2C- 3
4.
TÝch ph©n Q( x
) = ( x − a ) ( x 2 + px + q ) , ta t×m A, B, C sao cho : P( x ) A Bx + C = + 2 Q( x ) x − a x + px + q + NÕu Q(x) gåm nh©n tö bËc hai ®¬n vµ nh©n tö bËc hai béi: Q( x ) = ( x − a ) ( x 2 + px + q ) , ta t×m A, B 1 , C1 , B 2 , C 2 sao cho : 2 P( x ) A B x + C1 B x + C2 = + 2 1 + 22 Q( x ) x − a ( x + px + q ) x + px + q 2 Bµi tËp minh ho¹: 3 4x 2 + 16x − 8 2 3x 2 + 3x + 3 5 x+1 1. ∫ x 3 − 4x dx 2. ∫ 3 dx 3. ∫x − x2 dx 1 x − 3x + 2 3 2 2 IV. TÝch ph©n hµm v« tû ®¬n gi¶n: b b dx ∫ ∫ ax + b = ( ax + b ) n 1 1.D¹ng: n ax + b .dx; : §æi n a a n ax + b b 2.D¹ng: ∫ a ax 2 + bx + c .dx b - NÕu a>0 : TÝch ph©n cã d¹ng ∫ a u 2 + a 2 du ®Æt u=atgt HoÆc chøng minh ngîc c«ng thøc: u 2 2 u2 ∫ u + a du = 2 u + a + 2 ln u + u + a + C 2 2 2 2 b -- NÕu a<0 : TÝch ph©n cã d¹ng ∫ a a 2 −u 2 du ®Æt u=asint b dx 3.D¹ng: ∫ ax 2 + bx + c a dx 1 ( x − x 1 ) − ( x − x 2 ) dx - NÕu ∆ > 0 : ∫ = ∫ a( x − x )( x − x ) = ... a( x − x 1 )( x − x 2 ) x 2 − x 1 1 2 dx dx ∫ =∫ b = - NÕu ∆ = 0 : 2 b a x − a x − 2a 2a dx - NÕu ∆ < 0 : Víi a>o: ∫ §Æt ( x − α ) = β.tgt ( x − α) 2 + β 2 du HoÆc chøng minh ngîc c«ng thøc: ∫ = ln u + u 2 + a 2 + C u +a 2 2 dx Víi a<0: ∫ §Æt ( x − α ) = β. sin t β 2 − ( x − α) 2 Bµi tËp minh ho¹: -N2C- 4
5.
TÝch ph©n
3 dx 1 dx 1 dx 1. I = ∫ 2. I = ∫ 3. I=∫ 2 4. 0 x 2 − 3x + 2 0 x 2 + 2x + 1 0 x + x+1 1 dx I=∫ 0 − x 2 − 2x + 3 1 1 5. I = ∫ x + x + 1.dx 6. I = ∫ − x − 2x + 3.dx 2 2 0 0 b dx 1 4.D¹ng ∫ ( x + α) §Æt ( x + α ) = a ax + bx + c 2 t 1 dx 1 dx BTMH: 1. ∫ 2. ∫ ( x + 1) x + x + 1 0 2 ( 2x + 4) x + 2x 0 2 ∫ R ( ( ax + b ) ; ( ax + b ) ).dx §Æt t = ( ax + b ) m q p 1 5.D¹ng: n s víi s lµ BCNN cña n vµ q. 1 dx 1 dx 1 6 x BTMH: ∫ ∫ ∫ 1+ dx − ( 2x + 1) 0 3 ( 2x + 1) 2 0 ( 1 − 2x ) − 4 ( 1 − 2x ) 0 3 x V. TÝch ph©n hµm sè lîng gi¸c: b 1.D¹ng: ∫ f ( sin x; cos x ) dx a - NÕu f lµ hµm lÎ theo sinx: §Æt t=cosx. - NÕu f lµ hµm lÎ theo cosx: §Æt t=sinx. - NÕu f lµ hµm ch½n theo sinx vµ cosx: §Æt t=tgx. Bµi tËp minh ho¹: π π π π 1. ∫ sin x dx 2. 2 3 6 cos x 3 3. 4 dx 4. 4 dx 3 ∫ 4 + sin x dx ∫ sin x. cos 3 ∫ ( sin x + cos x ) 2 0 cos x 0 0 x 0 b 2.D¹ng: ∫ sin x. cos x.dx m n a - NÕu m vµ n ch½n: H¹ bËc. - NÕu m lÎ: §Æt t=cosx. - NÕu n lÎ: §Æt t=sinx. Bµi tËp minh ho¹: π π π π 1. ∫ sin 3 x. cos 2 x.dx 2. ∫ sin 4 x. cos 2 x.dx 3. ∫ sin x dx 4. dx 2 2 2 4 2 0 0 2 0 cos x ∫ cos 0 4 x. sin 4 x b 3.D¹ng: ∫ R ( sin x;cos x ) .dx trong ®ã R lµ hµm h÷u tØ theo sinx, cosx. a x 2dt 2t 1− t2 2t §Æt t = tg ⇒ dx = ; sin x = ; cos x = ; tgx = 2 1+ t2 1+ t2 1+ t2 1− t2 -N2C- 5
6.
TÝch ph©n
b dx Cô thÓ lµ hµm: I=∫ a a sin x + b cos x + c Bµi tËp minh ho¹: π π π 1. I = ∫ 4 dx 2. 2 ( 1 + sin x ) 3. I = ∫ dx 2 I=∫ dx 0 sin x + cos x + 1 0 sin x.( cos x + 1) 0 ( cos x + 2) b a sin x + b cos x 4.D¹ng: I=∫ dx a c sin x + d cos x Ph©n tÝch: (Tö sè)=A.(MÉu sè)+B.(MÉu sè)’ a sin x + b cos x b b b c cos x − d sin x b b d( c sin x + d cos x ) I=∫ dx = A ∫ dx + B.∫ dx = A ∫ dx + B.∫ a c sin x + d cos x a a c sin x + d cos x a a c sin x + d cos x π Bµi tËp minh ho¹: I = 3sin x − 2cos x dx 2 ∫ 4sin x + 3cos x 0 b a 1 sin x + b 1 cos x + c1 5.D¹ng: I = ∫ dx a a 2 sin x + b 2 cos x + c 2 Ph©n tÝch: (Tö sè)=A.(MÉu sè)+B.(MÉu sè)’ +C b b a 2 cos x − b 2 sin x b dx I = A ∫ dx + B ∫ dx + + C∫ a a a 2 sin x + b 2 cos x + c 2 a a 2 sin x + b 2 cos x + c 2 b d( a 2 sin x + b 2 cos x + c 2 ) b = A ∫ dx + B ∫ + C.J a a a 2 sin x + b 2 cos x + c 2 J lµ tÝch ph©n tÝnh ®îc. π π Bµi tËp minh ho¹: 1. I = sin x − cos x + 1 dx 2. I = sin x + 1 2 2 ∫ sin x + 2cos x + 3 0 ∫ 3sin x − 4cos x + 5 dx 0 VI. PhÐp ®æi biÕn ®Æc biÖt: b I = ∫ f ( x)dx a Khi sö dông c¸c c¸ch tÝnh tÝch ph©n mµ kh«ng tÝnh ®îc ta thö dïng phÐp ®æi biÕn: t = ( a + b ) − x .Thùc chÊt cña phÐp ®æi biÕn nµy lµ nhê tÝnh chÊt ch½n lÎ cña hµm sè f(x). Bµi tËp minh ho¹: π ( ) π 2 cos x 1 x sin x 1. I = ∫ x dx 2. I = ∫ ln 3 x + x 2 + 1 dx 3. I = ∫ dx 4. πe + 1 0 1 + cos x 2 −1 − 2 1 sin 2004x I= ∫ dx −1 2003 + 1 x Chøng minh r»ng: -N2C- 6
7.
TÝch ph©n 1. NÕu
f(x) lµ hµm sè ch½n vµ liªn tôc trªn [ − a; a] th×: a a ∫ f ( x)dx = 2.∫ f (x)dx −a 0 a 2. NÕu f(x) lµ hµm sè lÎ vµ liªn tôc trªn [ − a; a] th×: ∫ f ( x)dx = 0 −a π π π π 2 2 2 2 3. ∫ f (sin x )dx = ∫ f (cos x )dx 4. ∫ x.f (sin x)dx =π ∫ f (sin x)dx 0 0 0 0 -N2C- 7
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