基本积分公式表=∫kdx)1(kxC+)(是常数k=∫dxxμ)2(11++μμx,C+)1(?≠μ∫=xdx)3(||lnxC+=+∫dxx211)4(Cx+arctan=?∫dxx211)5(Cx+arcsin∫=xdxcos)6(Cx+sin∫=xdxsin)7(Cx+?cos∫=xdx2sec)8(Cx+tan∫=xdx2csc)9(Cx+?cot∫=xdxxtansec)10(Cx+sec∫=xdxxcotcsc)11(Cx+?csc=∫dxex)12(Cex+=∫dxax)13(Caax+ln第二节换元积分法（一）一、第一换元积分法问题∫dxex2?=,2数不是积分公式表上的函被积函数xe用直接积分法，求不出它的积分。怎么办？∫dxex2=∫xe2)2(xd21=21∫xe2)2(xd=xu2=21∫uedu=21ueC+=21xe2C+一般情况下：,)()(uFuf有原函数设)()('ufuF=即∴)(∫duuf=uF)(C+可导若)(xu?=)(uF=)]([xF?∴=)]([xFdxd?)(')('xuF?=)(uf)('x?=)]([xf?)('x?∴的原函数是)(')]([)]([xxfxF???∴dxxxf)(')]([??∫=)]([xF?C+=)(uFC+=∫duuf)(:,我们就得到下面的定理这样设)(uf具有原函数，)(xu?=可导，则有换元公式)(xu定理1?==dxxxf)(')]([??∫duuf)(:用法∫dxxg)(=∫)]([xf?)('x?dx=∫)]([xf?)]([xd?=∫)(ufdu)(xu?==)(uFC+=)]([xF?C+)(xu?=是积分公式表上的函数)(uf“凑”微分法例1求.2sin∫xdx解1∫xdx2sin=∫x2sin)2(xd21=∫x2sin)2(xd21=xu2=令∫usindu21=)cos(u?21C+=)]2cos([x?21C+=)2cos(x21?C+解2∫xdx2sin=∫xxcossin2dx=∫xsin)(sinxd2=xusin=令∫udu2=22u2C+=2uC+=x2sinC+解3∫xdx2sin=∫xxcossin2dx=∫xcos)(cosxd2=xucos=令∫udu2?=22u2?C+=2u?C+=x2cos?C+=?xxsincos2dx)1(?=∫xcos)(cosxd2?例2+dxx231=∫x231+)23(xd+=xu23+=令∫u1du21=||lnu21C+=|23|ln21x+C+21=∫x231+)23(xd+21例3?dxx)43(sec2=∫)43(sec2?x)43(?xd=43?=xu令∫u2secdu31=utan31C+=)43tan(31?xC+31=∫)43(sec2?x)43(?xd31例4?dxxx21=∫21x?)1(2xd?=21xu?=令∫u