积分考研十道典型例题

积分考研二十道典型例题不会的数学题可以发邮箱交流[e-mail]解题技巧总结①看不懂没事，跳过接着看②先用一般方法，再用特别方法③套公式，公式很重要，智慧的结晶④三角函数和一般函数相乘，用分部积分法⑤出现三角函数，如tanx，就可以画三角形了⑥每种类型的简单的公式要记⑦数学整的是想法，而不是难题⑧凡是比较难的，比较牛的都是可以带公式的⑨学习只有两个境界，看和思，达到第一个就已经是高手了⑩极其重要的公式①dxxxd211arctan②Cxxxdxxdtanln2sin222csc③dxvuuvdxuv''④Cxaxxadx2222ln⑤Cxadxxax2222⑥Cxudxxuu111⑦Cbaxabaxdxln1⑧Cxxdxcoslntan⑨xdxxd2sectan⑩Cbxbbxaebabxdxeaxaxcossin1sin22110arctan12bCxbaabaxbdx12Cxxdxarctan1213dxxxd211arcsin14①分项积分法Cxdxdxdxxd34373134333x-11213x-1271313x-191313x-19131311-3x-191313x-1x②换元积分法CxxxCxxxxxxdxxxdxxdxxxxlnln111lnlnln1ln1lnln12222③分部积分法公式：dxvuuvdxuv''Cexeexexxdeexexdexexdexdxdexdxexxxxxxxxxxxxx663633232323333④有理数的积分Cxxdxxxxxdxxxxxdxxxxdxxxxln8ln9288291888888999999998999109⑤三角有理式的积分Cxxdxxcoxdxxxxdxxxdx22tanarctan212tan2tan2122tan222sin212coscossin2222244⑥无理式的积分CtttCtttdtttttdtttdtttxxdxxxx11lnarctan211lnarctan21111211t414dxt11111222222222原式，令⑦杂题CxeCetdtdttdtdxtdxexttttx1212e2te2te22,x原式令⑧真分式的积分CxxdxxxdxxxxBABABABAxBAxxBxAxxBxAxxxdxxxx2ln33ln4233465143123123123132651651222解：⑨可化为有理数的积分C2tanln212tan2tan41ln214112212121111212121cos1sinsin112,arctan211cos,12sin,2tanucos1sinsin1222222222222xxxCuuuduuuduuuuuuuuuduuudxxxxduudxuxuuxuuxxdxxxx解：⑩积分表的使用CxxxxCxxxxxxdxxxdxx42sin2434cossin22cossin434cossinsin434cossinsin3323411(20120406)(答案待定154)CxxxCtttdttdttdttttdxxxx2secln4142tan1ln211ln21arctan1ln21t111121t11t1121dtt112tt2xtancossin1sin222222原式令解：12（20120407）Ce1ln11111xxeedxdxeeeedxxxxxxx13（20120407129）Cxxxxdxxddxxxdxxxdxxxxxxxdxtanlncos212sin2coscoscossin1cossincossincossincossin233322314（20120407129）Cxxxdxdxxdxxxdxxxxdxxxxxdxxx323323223222232222264arctan31arctan13111111111115（20120407133）CxexeCxexexedxexexexexexexeddxxexeexdxxexxxxxxxxxxxxxxxxxx1ln1lnln111111116（20120407141）CexCexexdexexdexxdexdxxexdxexxxxxxxxxsinsinsinsinsinsinsinsinsin1sin22sin2sin2sin2sin2sinsin2cossin22sin17（20120407140）Cxxxxxxdxxxdxxxxxxdxxx2222222211ln11211ln11ln1ln18（20120407141）Cxxxdxdxxxxxxxdxxdxdxxxlnln1ln1lnlnlnln1ln222219（201204070408141142）最后一步待定，已定CxxCxxxxCxxxxxdxxxxxxxdxxddxxxdxxxdxxxdxxxx2tan2cosln2cosln2tancos1ln2cosln22tancos1ln2tan2tancos1ln2tancos1cos2cos2cos1sincos1cos1sin22220（20120408143）CxxxeCxxxeCuueudeudeudeuuduuuueuduecsuuexuududxuxdxxxexxuuuuuux22arctan22arctan242222222arctan15112221012cos22sin1012sin21cossinsectansecsectantan1tanarctan,sec,tan1原式令解：21（20120408144）Cxxcoxxxdxxdxxxxcoxxxdxxxxxxdxxdxxlnlnsin21lnsinlnsinlnlnsinlncoslnsinlnsinlnsinlnsin22（20120408149）Cxxxdxdxxdx2arctan21tan2tansec1coscos1222223（20120408139）221arcsinarcsin1arcsin11arcsinarcsinxdxxxxdxxxxxddxxx