Chuyên đề Hệ thống Nguyên hàm Tích phân

<span class='text_page_counter'>(1)</span>I - TÝch ph©n c¸c hµm ®a thøc, hµm sè luü thõa b. u 1  Chó ý :  u du  víi   0 vµ -1,  1 a a b. n 5. m n. x 5. b 1 du  ln u ; ab  0. a u a. 1  u  n , du = u’(x)dx n u. u  u ; u  0, n  N * , m. b. 5. I1 =  (3  )4 dx. I5 =. I2 =  x5 (1  x3 )6 dx. I6 =. I3 =  x(1  x)11dx. I7 =  ( x x  3 )dx. 0 1. 0 1. . I9 =. x  3 dx. 2 2. x. 2.  2 x  3 dx. I4 =  x(1  x ) dx. 1 x. 2 I8 =  1  2 x3  x dx. . 1 3. 2. I11 =  max 3x  2; x 2 dx. x. 1. 0. 3. 2 2   I10 =  x 1  x 3  dx   0.125 1. 1 8. 4. 2 n. 1 dx x 1  x 1.  1. 1. 0 1. 3. 0. II- TÝch ph©n c¸c hµm h÷u tØ. I12 =. 2. 4 1 (3  2 x)4 dx 1. 2x 1 I13 =  dx x  2 1. 1. x3  x  1 0 x  1 dx 2. x dx I16 =  3 ( 3 x  1 ) 0 b. 1 dx I17 =  ( x  a )( x  b) a 2. 1 dx I18 =  2 0 x  2x  2. I19 =. 1.  0. 4 x  11 dx 2 x  5x  6. 4. I21 =  x 2 dx ( x  2) 1. 2x  2  I14 =    3 dx x  1   0. 1. 1. 1. 1. 1. I15 =. 3. I30 = . 1 I20 =  3 dx 2 2 x  2x  x. I31 =. I22 =  x 2  7 x  3dx x  5x  4 2. 2. x 1  I23 =    dx x2 2. I32 =. dx I25 =  dx 3  x2 1 0. I27 =. 1.  0. I28 =. 1  x . 4x  2.  ( x  2)( x. 2.  1). dx. 2. 0. 3. 2 2. 1. b. 2. x3. (x  x ) dx x4. I33 =  a  x2 2 dx (a  x ). 1. 1. . 1 3 3. 0. I24 =  2 2 x  5 dx x  4x  7. I26 = .  3. 1. 2. 4. 1 dx x  x3. 1. 0. dx. I35 =. x.  (1  x. I34 =. 2 3. ). dx. 2. 1 x2 1 1  x 4 dx 1. x2. 1  x  2. 1 x4 dx I36 =  6 0 1 x. dx 3. 1. 3 0 1  x3 dx 1. I29 =  12 1  1  x  x. I37 =. 1 5 2.  1. 2009. dx. 1 2. x2  1 dx x4  x2  1. 5. x2 1 I38 =  2 dx ( x  5 x  1)( x 2  3x  1) 3. III- TÝch ph©n hµm chøa c¨n thøc b. Chú ý:  R( x, f ( x))dx Trong đó R(x, f(x)) có các dạng: a. +) R(x,. ax  ) §Æt x = a cos2t, t  [0; ] ax 2. +) R(x,. a 2  x 2 ) §Æt x = a sin t hoÆc x = a cos t. -------------------------------------------------------------------------------------------------------Lop12.net.

<span class='text_page_counter'>(2)</span> +) R(x,. n. ax  b ) §Æt t = cx  d. +) R(x, f(x)) =. ax  b cx  d. n. 1. Víi ( x 2  x   )’ = k(ax+b). (ax  b) x 2  x  . Khi đó đặt t = x 2  x   , hoặc đặt t = +) R(x,. a 2  x 2 ) §Æt x = a tgt , t  [. +) R(x,. x 2  a 2 ) §Æt x =. +) R. I39 =. . n1. n2. ni. . cos x. I53 =. 3  2xdx. 3 1. I54 =. 0 1. x. 3. 2. I56 =. 1 dx x2. I43 =  1 1. I47 = I48 =. 4.  0 2. I58 =. . x2 dx 3 x 1.  0. x 1 dx x 1. 2. 3. x 3. 1  x2. 2. 1. . x x2 1. 2 3. 2 3. I50 = . 5. I61 =. 0. dx. I62 =. dx. I63 =. x2 1. 10  x 2 dx. 0 1. . dx x3  1. 1. ln 3. I67 =. x. 2. 2 x. 2. dx. 3. . 7. . I68 = I69 =. 2 3. . I64 =. ln 2. . 0 ln 2.  1. 1 1  ex. dx. e2 x. dx 1  ex ln 2 x dx x 1  ln x.  2. I70 =  0. cos xdx 7  cos 2 x.  2. I71 =  sin x. cos x  cos 2 x dx. 0. . dx. 3. (1  x ). 2 3. 1 1  x2. I72 =  0. dx. 1 x 3 2.  0. 4  x 2 dx. cos xdx 2  cos 2 x.  2. dx. x  1dx 2. 2. I73 =  6 1  cos 3 x sin x cos 5 xdx 0. . I74 =. 2.  0. 1. .  x 2  2 x  3dx. 1. 2 x dx 2 x. sin 2 x  sin x 1  3 cos x. dx. 0. I65 = . 3. 1 x2. . 0. 0. x. 1 2. 0. x x2  4. I52 =  x. x. I60 = . dx. 5. 2. 2 2. x 3  2 x 2  x dx 3. x 3 dx. 1  3ln x ln xdx x. I66 = . 0. 4. I51 = . dx. I59 =  x 2 1  x 2 dx. 2. .  x. 0 1. x 1 1  x  1dx. 0. I49 =. I57 =. x dx 2x 1. 0 3. 1 x. 2. 0. 0. I46 =. 1. §Æt x = tk e. x.  x. I55 =  x 3. 1  x 2 dx. 2. 7. 1. 0. I42 =  3x 2 4 1  x3 dx. I45 =. 2. 3. 1. I44 =.  , t  [0;  ] \ { }. 0. I40 =  x 1  xdx I41 =. ; ] 2 2. x ; x ;...; x Gäi k = BCNH(n1; n2; ...; ni),. 1. . a.  . 1 ax  b. dx. 1.  3. I75 =  0. cos 2 x  2 3tgx cos 2 x dx cos 2 x. IV- Tích phân hàm số lượng giác. Chú ý: Các công thức lượng giác. -------------------------------------------------------------------------------------------------------Lop12.net.

<span class='text_page_counter'>(3)</span> TÝch thµnh tæng : 2sinax.cosbx = sin(a+b)x + cos(a-b)x 2cosax.cosbx = cos(a+b)x + cos(a-b)x 2sinax.sinbx = cos(a-b)x – cos(a+b)x H¹ bËc: 2sinax.cosax = sin2ax; 2sin2ax =1- cos2ax; 2cos2ax = 1+ cos2ax. x 2. BiÓu diÔn theo t = tan ; sinx =. 1 t2 2t 2t ; cosx = ; tanx = 2 2 1 t 1 t 1 t2. C¸c vi ph©n: d(sinx) = cosxdx; d(cosx) = -sinxdx; d(tanx) = . I76 =. 2. 2. x cos 4 xdx. I88 =. 0. 2 3  sin x cos xdx. . I78 =  14 dx cos x 0. 4. 0. 6. 4. 6. xdx. I80 =  cos 2 x(sin 4 x  cos 4 x)dx 0. . I92 =. I93 =. sin 3 x 0 1  cos 2 x dx . 6  4.  sin 0. 2. dx x  2 sin x cos x  cos 2 x. . cos 3 x I85 =  dx 1  cos x 0. I96 =. 2. cos xdx  (1  cos x) 2 3. 4. I104 =. sin x  7 cos x  6.  4 sin x  5 cos x  5 dx. I105 = I106 =. 1  cos 2 x  sin 2 x dx sin x  cos x 0. 4 sin xdx.  (sin x  cos x). 3. 4. sin 3 x sin 4 x. dx   tgx  cot g 2 x 2.  sin. 2. sin 2 xdx x  5 sin x  6. 4. dx.  (sin x  2 cos x). 2. 0. . I107 =. sin 3 x. 2.  . 1 dx 3 sin x  cos x. 6 .  1  cos x dx 0. I108 =. dx.   sin 2 x  sin x. I109 =. 4. . I98 =. 3. . . 2.  tgxtg ( x  6 )dx . 0. 4 sin 3 x 0 1  cos 4 x dx. . I97 =. . 3. 6 . 4. 2. ). . ). . 4 6 I86 = tan x dx. . . 2. I95 =. 6. 0. . . 0. 2. I94 =. 2.  cos 2 x. I103 =. dx cos x cos( x . . 6. 0 . 3. dx I83 =  4  sin x. cos x. . sin x sin( x . . . 2. I82 =. 1 0 1  tgx dx. 0. 1 I81 =  sin x dx . I102 =. 4. 4. dx. . xdx. . 2. 3. 3. . I91 =.  2. 3. . 2. 0. I101 =.  . .  sin. sin 3 x  sin x dx sin 3 xtgx. .  cot g . 5. . . 2. I89 =  tg 3 xdx I90 =. 3 3. I100 =. 4. 4. I87 =. . sin x  cos x  1 dx sin x  2 cos x  3. . 2. 0. I84 =. 2. . . I79 =. . .  sin. I77 =. dx =(1+tan2x)dx. cos 2 x. sin 3 x 0 cos 2 x dx 4. 4. sin 4 xdx 6 x  cos 6 x.  sin 0  3.  . tan 2 x  cot 2 x  2dx. 6.  3.  2. I99 =  sin 2 x(1  sin 2 x) 3 dx. I110 =  sin 2 x tan xdx 0. 0. V- TÝch ph©n tæng hîp c¸c hµm sè b. b. Chó ý : C«ng thøc tÝch ph©n tõng phÇn:  udv  uv a   vdu b. a. a. -------------------------------------------------------------------------------------------------------Lop12.net.

<span class='text_page_counter'>(4)</span> 1. 1. I111 =  xe2 x dx 0. 1. 1. I112 =  (2  x) s inxdx 0. xdx. I115 =  x ln xdx. . I116 =  ln( x  x)dx. I120 =. 2. . e. 6. . I124 =  ln(1  cot x)dx .  cos(ln x)dx 1. ln(1  x) I117 =  dx x2 1. 3. I123 =  ln(sin2 x) dx cos x. 0. 3. e x  1e 2 x dx. ln 3. I119 =  ln(1  tan x)dx. 1. 2. 0. I122 = . 4. 2. 2. ln 8. x3  1 ln xdx x. . 2. 2.  sin. I118 = . e. . I113 =. e3. I114 =  ( x 2  1)e x dx. 2. 4. 0. I121=  x(e x . x  1)dx. 1. VI – Một số tích phân đặc biệt . 1. I125.  ln( x  1  x 2 )dx 1. I129.. I127..  2.  . . x  cos x dx 4  sin 2 x. 2. 3. 2 I128. x  1x dx  31  2. . . 1. x 4  sin x I126  dx 2 1 1  x. 2. . sin x sin 3 x cos 5 x dx 1 ex. 1. 2.  0. sin x sin x  cos x. dx. dx 2 1 (1  e )(1  x ). I134. . . . I131.  x sin x dx 2  cos x 0. I135.. x. sin 5 x. 2.  . . . 2. 1  cos x. dx. x sin x dx 2 0 1  cos x. I132. . a. CMR Hµm sè f(x) liªn tôc trªn [-a; a], th×. . a. liªn tôc trªn [-. 0. 2. . I130.. 4. I133.  sin 4 x ln(1  tgx)dx. a. f ( x)dx   [ f ( x)  f ( x)]dx. ¸p dông cho f(x). 0. 3 3 ; ] tháa m·n f(x) + f(-x) = 2 2. 2  2 cos 2 x , TÝnh: I 136=. 3 2. .  f ( x)dx . 3 2. -------------------------------------------------------------------------------------------------------Lop12.net.

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