不定积分 1 1 dx=2(2x+3) d(sinx)=cosxdx d(Inx )=dx dx=d(3x+5) d(cosx )sinxdx d(ex )=exdx dx=d(5x-5) d(tanx )sec2xdx 1 d(2x)=后dx dx=d(4-5x) Vx d(cotx )csc2xdx 1 d(-)= xdx =d( X xzdx d(secx )secxtanxdx 1 x2dx =d( d(arctanx)=1+xidx d(-cscx)=cscxcotxdx 3xdx d( 1 durcsinx)=dx
𝒅𝒙 = 𝟏 𝟐 𝒅(𝟐𝒙 + 𝟑) 𝒅𝒙 =_𝒅(𝟑𝒙 + 𝟓) 𝒅𝒙 =_𝒅(𝟓𝒙 − 𝟓) 𝒅𝒙 =_𝒅(𝟒 − 𝟓𝒙) 𝒙𝒅𝒙 = 𝒅( ) 𝒙 𝟐𝒅𝒙 = 𝒅( ) 𝟑 𝒙𝒅𝒙 = 𝒅( ) 𝒅( ) = 𝒄𝒐𝒔𝒙𝒅𝒙 𝒅( ) = 𝒔𝒊𝒏𝒙𝒅𝒙 𝒅( ) = 𝒔𝒆𝒄𝟐𝒙𝒅𝒙 𝒅( ) = 𝒄𝒔𝒄𝟐𝒙𝒅𝒙 𝒅( ) = 𝒔𝒆𝒄𝒙𝒕𝒂𝒏𝒙𝒅𝒙 𝒅( ) = 𝒄𝒔𝒄𝒙𝒄𝒐𝒕𝒙𝒅𝒙 𝒔𝒊𝒏𝒙 −𝒄𝒐𝒔𝒙 𝒕𝒂𝒏𝒙 −𝒄𝒐𝒕𝒙 𝒔𝒆𝒄𝒙 −𝒄𝒔𝒄𝒙 𝒅( ) = 𝒆 𝒙𝒅𝒙 𝒅( ) = 𝟏 𝒙 𝒅𝒙 𝒆 𝒙 𝒍𝒏|𝒙| 𝒅( ) = 𝟏 𝒙 2 𝒙 𝒅𝒙 𝒅( ) = 𝟏 𝒙 𝟐 − 𝒅𝒙 1 𝑥 𝒅( ) = 𝟏 𝟏 − 𝒙 𝟐 𝒂𝒓𝒄𝒔𝒊𝒏𝒙 𝒅𝒙 𝒅( ) = 𝟏 𝟏 + 𝒙 𝟐 𝒂𝒓𝒄𝒕𝒂𝒏𝒙 𝒅𝒙
第三讲 换元积分法
第三讲 换元积分法
不定积分 一基本积分公式 (8) sinxdx =-cosx +c (1) kdx=kx+c (9) cosxdx sinx +c xu+1 (2) xudx u+1 +c(u≠-1) (10) sec2xdx tanx +c (3) -dx =Inx+c (11) csc2xdx =-cotx +c (4) exdx =ex+c (12) secxtanxdx secx +c Q (5) axdx -+c(a>0,a≠1) (13) cscxcotxdx =-cScx +c In 1 (6) x dx=-+c X (14) 1 V1-x2 dx arcsinx +c dx=2vx+c (15) 1+x2 dx arctanx +c
(2) න 𝑥 𝑢𝑑𝑥 = 𝑥 𝑢+1 𝑢 + 1 + 𝑐 (𝑢 ≠ −1) (1) න 𝑘𝑑𝑥 = 𝑘𝑥 + 𝑐 (3) න 1 𝑥 𝑑𝑥 = 𝑙𝑛 |𝑥| + 𝑐 (4) න 𝑒 𝑥𝑑𝑥 = 𝑒 𝑥 + 𝑐 (5) න 𝑎 𝑥𝑑𝑥 = 𝑎 𝑥 𝑙𝑛𝑎 + 𝑐 (𝑎 > 0, 𝑎 ≠ 1) (6) න 1 𝑥 2 𝑑𝑥 = − 1 𝑥 + 𝑐 (9) න 𝑐𝑜𝑠𝑥𝑑𝑥 = 𝑠𝑖𝑛𝑥 + 𝑐 (8) න 𝑠𝑖𝑛𝑥𝑑𝑥 = −𝑐𝑜𝑠𝑥 + 𝑐 (10) න 𝑠𝑒𝑐2𝑥𝑑𝑥 = 𝑡𝑎𝑛𝑥 + 𝑐 (11) න 𝑐𝑠𝑐 2𝑥𝑑𝑥 = −𝑐𝑜𝑡𝑥 + 𝑐 (12) න 𝑠𝑒𝑐𝑥𝑡𝑎𝑛𝑥𝑑𝑥 = se𝑐𝑥 + 𝑐 (13) න 𝑐𝑠𝑐𝑥𝑐𝑜𝑡𝑥𝑑𝑥 = −𝑐𝑠𝑐𝑥 + 𝑐 (14) න 1 1 − 𝑥 2 𝑑𝑥 = 𝑎𝑟𝑐𝑠𝑖𝑛𝑥 + 𝑐 (15) න 1 1 + 𝑥 2 𝑑𝑥 = 𝑎𝑟𝑐𝑡𝑎𝑛𝑥 + 𝑐 (7) න 1 𝑥 𝑑𝑥 = 2 𝑥 + 𝑐 一.基本积分公式
不定积分 练习：求下列不定积分。 (1)∫x6dx (2)∫x-8dx (3)∫2xdx 解：= x6+1 6+i+C 解：=t+C -8+1 解：= +c 号+c + 7+C
�� (1( �� (2𝟔𝒅𝒙 ( �� (3−𝟖𝒅𝒙 ( 𝒙𝒅𝒙 = 𝒙 𝟔+𝟏 𝟔+𝟏 + 𝑪 = 𝒙 𝟕 𝟕 + 𝑪 解： 解：= 𝒙 −𝟖+𝟏 −𝟖+𝟏 + 𝑪 = 𝒙 −𝟕 −𝟕 + 𝑪 解：= 𝟐 𝒙 𝒍𝒏𝟐 + 𝑪 练习：求下列不定积分
不定积分 练习：求下列不定积分。 (1)∫(5x+2x+sinx)dx(2)∫(7x+x10+3)dx 影+2-cosx+G =++3x+c (3)∫(2ex+5cosx)dx (4)(径-c0sx+3x2+2dx =2ex+5sinx C =2mx-sinx+x3+器+C
练习：求下列不定积分。 ��) (1( ��) (2𝒙 + 𝟐𝒙 + 𝒔𝒊𝒏𝒙）𝒅𝒙 ( 𝒙+𝒙 𝟏𝟎 + 𝟑)𝒅𝒙 �𝟐�) (3( ) (4𝒙 + 𝟓𝒄𝒐𝒔𝒙)𝒅𝒙 ( 𝟐 𝒙 − 𝒄𝒐𝒔𝒙 + 𝟑𝒙 𝟐 + 𝟐 𝒙 )𝒅𝒙 = 𝟓 𝒙 𝒍𝒏𝟓 + 𝒙 𝟐 − 𝒄𝒐𝒔𝒙 + 𝑪 = 𝟕 𝒙 𝒍𝒏𝟕 + 𝒙 𝟏𝟏 𝟏𝟏 + 𝟑𝒙 +C = 𝟐𝒆 𝒙 + 𝟓𝒔𝒊𝒏𝒙 + 𝑪 = 𝟐 𝒍𝒏 𝒙 − 𝒔𝒊𝒏𝒙 + 𝒙 𝟑 + 𝟐 𝒙 𝒍𝒏𝟐 + 𝑪