Chuyên đề Hệ thống Nguyên hàm Tích phân

IV- Tích phân hàm số lượng giác.. Chú ý: Các công thức lượng giác..[r]

I - TÝch ph©n c¸c hµm ®a thøc, hµm sè luü thõa b u 1  Chó ý :  u du  víi   vµ -1,  1 a a b n m n x b du  ln u ; ab  a u a  u  n , du = u’(x)dx n u u  u ; u  0, n  N * , m b I1 =  (3  )4 dx I5 = I2 =  x5 (1  x3 )6 dx I6 = I3 =  x(1  x)11dx I7 =  ( x x  )dx 1  I9 = x  dx 2 x  x  dx I4 =  x(1  x ) dx x I8 =   x3  x dx  I11 =  max 3x  2; x dx x 2   I10 =  x 1  x  dx   0.125 1 n dx x 1  x 1  1 II- TÝch ph©n c¸c hµm h÷u tØ I12 = 1 (3  x)4 dx 2x 1 I13 =  dx x  1 x3  x  0 x  dx x dx I16 =  ( x  ) b dx I17 =  ( x  a )( x  b) a dx I18 =  x  2x  I19 =  x  11 dx x  5x  I21 =  x dx ( x  2) 2x   I14 =    dx x    1 1 I15 = I30 =  I20 =  dx 2 x  2x  x I31 = I22 =  x  x  3dx x  5x  2 x 1  I23 =    dx x2 2 I32 = dx I25 =  dx  x2 I27 =  I28 = 1  x  4x   ( x  2)( x  1) dx 2 b 2 x3 (x  x ) dx x4 I33 =  a  x2 dx (a  x ) 1  3 I24 =  2 x  dx x  4x  I26 =   1 dx x  x3 dx I35 = x  (1  x I34 = ) dx 1 x2 1  x dx x2 1  x  1 x4 dx I36 =  1 x dx 3 0  x3 dx I29 =  12 1   x  x I37 = 1  2009 dx x2  dx x4  x2  x2 1 I38 =  dx ( x  x  1)( x  3x  1) 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 +) R(x, a  x ) §Æt x = a sin t hoÆc x = a cos t -Lop12.net +) R(x, n ax  b ) §Æt t = cx  d +) R(x, f(x)) = ax  b cx  d n Víi ( x  x   )’ = k(ax+b) (ax  b) x  x   Khi đó đặt t = x  x   , đặt t = +) R(x, a  x ) §Æt x = a tgt , t  [ +) R(x, x  a ) §Æt x = +) R I39 =  n1 n2 ni  cos x I53 =  2xdx 3 I54 = x I56 = dx x2 I43 =  1 1 I47 = I48 =  I58 =  x2 dx x 1  x 1 dx x 1 x  x2  x x2 1 3 I50 =  I61 = dx I62 = dx I63 = x2 1 10  x dx  dx x3  1 ln I67 = x 2 x dx   I68 = I69 =  I64 = ln  ln  1  ex dx e2 x dx  ex ln x dx x  ln x  I70 =  cos xdx  cos x  I71 =  sin x cos x  cos x dx  dx (1  x ) 1  x2 I72 =  dx x 3   x dx cos xdx  cos x  dx x  1dx 2 I73 =   cos x sin x cos xdx  I74 =    x  x  3dx 2 x dx 2 x sin x  sin x  cos x dx I65 =  1 x2  0 x x x2  I52 =  x x I60 =  dx 2 x  x  x dx x dx  3ln x ln xdx x I66 =  I51 =  dx I59 =  x  x dx   x x 1  x  1dx I49 = I57 = x dx 2x 1 1 x 0 I46 = §Æt x = tk e x  x I55 =  x  x dx I42 =  3x  x3 dx I45 = 1 I44 =  , t  [0;  ] \ { } I40 =  x  xdx I41 = ; ] 2 x ; x ; ; x Gäi k = BCNH(n1; n2; ; ni),  a   ax  b dx 1  I75 =  cos x  3tgx cos x dx cos x IV- Tích phân hàm số lượng giác Chú ý: Các công thức lượng giác -Lop12.net 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 BiÓu diÔn theo t = tan ; sinx = 1 t2 2t 2t ; cosx = ; tanx = 2 1 t 1 t 1 t2 C¸c vi ph©n: d(sinx) = cosxdx; d(cosx) = -sinxdx; d(tanx) =  I76 = 2 x cos xdx I88 =  sin x cos xdx  I78 =  14 dx cos x 6 xdx I80 =  cos x(sin x  cos x)dx  I92 = I93 = sin x 0  cos x dx    sin dx x  sin x cos x  cos x  cos x I85 =  dx  cos x I96 = cos xdx  (1  cos x) I104 = sin x  cos x   sin x  cos x  dx I105 = I106 =  cos x  sin x dx sin x  cos x sin xdx  (sin x  cos x) sin x sin x dx   tgx  cot g x  sin sin xdx x  sin x  dx  (sin x  cos x)  I107 = sin x   dx sin x  cos x    cos x dx I108 = dx   sin x  sin x I109 =  I98 =    tgxtg ( x  )dx  sin x 0  cos x dx  I97 =   )  )  I86 = tan x dx   I95 =   I94 =  cos x I103 = dx cos x cos( x    dx I83 =   sin x cos x  sin x sin( x    I82 = 0  tgx dx I81 =  sin x dx  I102 = 4 dx  xdx  3  I91 =   I101 =     sin sin x  sin x dx sin xtgx   cot g    I89 =  tg xdx I90 = 3 I100 = 4 I87 =  sin x  cos x  dx sin x  cos x   I84 =   I79 =    sin I77 = dx =(1+tan2x)dx cos x sin x 0 cos x dx 4 sin xdx x  cos x  sin    tan x  cot x  2dx   I99 =  sin x(1  sin x) dx I110 =  sin x tan xdx 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 1 I111 =  xe2 x dx 1 I112 =  (2  x) s inxdx xdx I115 =  x ln xdx  I116 =  ln( x  x)dx I120 =  e  I124 =  ln(1  cot x)dx   cos(ln x)dx ln(1  x) I117 =  dx x2 I123 =  ln(sin2 x) dx cos x e x  1e x dx ln I119 =  ln(1  tan x)dx I122 =  2 ln x3  ln xdx x  2  sin I118 =  e  I113 = e3 I114 =  ( x  1)e x dx I121=  x(e x  x  1)dx 1 VI – Một số tích phân đặc biệt  I125  ln( x   x )dx 1 I129 I127     x  cos x dx  sin x I128 x  1x dx  31    x  sin x I126  dx 1  x  sin x sin x cos x dx 1 ex  sin x sin x  cos x dx dx 1 (1  e )(1  x ) I134    I131  x sin x dx  cos x I135 x sin x      cos x dx x sin x dx  cos x I132  a CMR Hµm sè f(x) liªn tôc trªn [-a; a], th×  a liªn tôc trªn [-  I130 I133  sin x ln(1  tgx)dx a f ( x)dx   [ f ( x)  f ( x)]dx ¸p dông cho f(x) 3 3 ; ] tháa m·n f(x) + f(-x) = 2  cos x , TÝnh: I 136= 3   f ( x)dx -Lop12.net