高等数学换元法.ppt.ppt

?教学目的:不定积分换元法?教学重点:凑微分法?教学难点:第二类换元法第二讲换元法主视图换元法凑微分法第二类换元法一次式的有理根式二次式的二次根式凑微分公式问题?xdx2cos,2sin21Cx??解决方法利用复合函数,?,21dtdx???xdx2cosdtt??cos21Ct????换元换元以后再还原求导数验证结果凑微分法设)(uf具有原函数,???dxxxf)()]([????)(])([xuduuf?第一类换元公式(凑微分法)说明使用此公式的关键在于将凑??dxxg)(.)()()]([)(????????duufdxxxfxu)(xu??可导,则有换元公式定理1难易凑微分法证明证由复合函数求导法则有可见)]([xF?是)()]([xxf???的一个原函数,故公式(1)(1)说明:当积分不便计算时,可考虑将g(x)化为的形式,那么????????duufxdxfdxxxfdxxg)()()]([)()]([)(????(2)对u积分求出)(uf的原函数)(uF,再以)(xu??代回即得所求积分,这种方法称为凑微分法.??)()]([')]([xxfxF?????)()]([xxf????dxxg)(?xdx解(一)?xdx2sin??)2(2sin21xxd;2cos21Cx???解(二)?xdx2sin??xdxxcossin2??)(sinsin2xxd??;sin2Cx??解(三)?xdx2sin??xdxxcossin2???)(coscos2xxd??.cos2Cx?????解,)23(23121231???????xxxdxx??231dxxx)23(23121??????duu??121Cu??ln21.)23ln(21Cx?????dxbaxf)(????baxuduufa])([1一般地例题例3求.)ln21(1dxxx??解dxxx??)ln21(1)(lnln211xdx???)ln21(ln21121xdx????xuln21????duu121Cu??ln21.)ln21ln(21Cx???例题例4求.)1(3dxxx??解dxxx??3)1(dxxx?????3)1(11)1(])1(1)1(1[32xdxx??????221)1(2111CxCx???????.)1(21112Cxx????????解dxxa??221dxaxa???222111?????????????????例题