20xx陳文燈考研數(shù)學(xué)復(fù)習(xí)指南習(xí)題詳解(理工)-高等數(shù)學(xué)ch3-文庫(kù)吧資料
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【正文】 t a n)0( 222 = cxaaax ??? a r c c o s22 8 3. dxeee xxx??? 21 )1( 解 . ???? deee xxx 21 )1( ?? dxee xx21+ dxee xx?? 221 =?? xxede 21－ dxeed xx??? 221 )1(21= cee xx ??? 21a r c s in 4. ?? dxxaxx 2 (a 0) 解 . ?? dxxaxx 2 xu?令 ?? duuau 2422 tau sin2?令 ? tdta 42 sin8 = ? ? ???? dtttadtta )2co s2co s21(24 )2co s1(8 2222 = ctatatadttatata ??????? ? 4s i n42s i n232 4c o s122s i n22 422222 = ctttattata ???? )s i n21(c o ss i nc o ss i n43 2222 = cttattata ??? c oss in2c oss in33 3222 = ca xaaxaxaa xaaxaaxa ????? 2222222232a r c s i n3 222 = cxaxxaaxa ???? )2(232a r c s in3 2 十 . 求下列不定積分 : 1. ? ?? dxxxcos2 sin2 解 . ??? ??????? xxddxxdxxx c o s2 )c o s2(c o s2 12c o s2 s i n2 tx ?2tan令 ?? ?????????? |c o s2|ln322|c o s2|ln112122 2222 xtdtxtttdt = cxxcxt ??????? |co s2|ln)2( t an31ar c t an34|co s2|ln3ar c t an34 9 2. ? ? dxxx xx c ossin c ossin 解 . ?? ? ???? dxxx xxdxxx xx c o ss i n 1c o ss i n2121c o ss i n c o ss i n = ? ? ? ????? ?? dxxxdxxxdxxxx co ss i n 121)co s( s i n21co ss i n 1co s )( s i n21 2 = ?????)4s in ()4(42)c o s( s in21??xxdxx = cxxx ???? |)82t a n (|ln42)c o s( s in21 ? 十一 . 求下列不定積分 : 1. ? ?? dxxxx )32(3 32 解 . ?? ????? ??? cxddxx xxxxxx 3ln3)3(3)32(3 3233 222 2. ? ??? dxxxx )13()523( 232 解 . )523()523(21)13()523( 2232232 ???????? ?? xxdxxdxxxx cxx ???? 252 )523(51 3. dxx xx? ? ?? 221 )1ln ( 解 . ?? ?????????? ?? cxxxxdxxdxx xx )1(ln21)1l n ()1l n (1 )1l n ( 222222 4. ?????? )11ln()11( 222 xxx x dx 解 . cxxxdxxxx d x ?????????????? ?? |)11l n (|ln)11l n ()11l n ()11l n ()11( 222222 十二 . 求下列不定積分 : 1. ?? dxx xx )1(arctan2 10 解 . ? ?? ???????? 1222222 )1(a r c t a n21)1()1(a r c t a n21)1( a r c t a n xxdxdx xdxx xx ? ??????????? dxxx xxdxx x 22222 )1( 1211a r c t a n21a r c t a n1 1211a r c t a n21 dttx xt d tx xtx ?? ????????? 2 2c o s1211a r c t a n21c o s211a r c t a n21t a n 222令 cttxx xa e xcttx x ???????????? c o ss i n41a r c t a n411 t a n212s i n81411a r c t a n2122 cxxxx xa e x ???????22 141a r c t a n411 t a n21 2. ?? dxxx1arcsin 解 . 令 txtxx 2t a n,1a r c s in ??? 則 ??? ???????? cttttt d ttttdtdxxx t a nt a nt a nt a nt a n1a r c s i n 2222 cxxxxcxxxxxx ??????????? 1a r c s i n)1(1a r c s i n1a r c s i n 3. ???? dxxxx x 222 11a r c s in 解 . ?? ? ???????? dtttt dtt ttttxdxxxx x )1( c s cc osc oss i n1s i ns i n11a r c s i n 222222 令 ? ?? ???????? ctt d tttdttt d tt 221c o tc o tc o t ctttt ????? 221|s in|lnc o t cxxx xx ?????? 22 )( a r c s i n21||ln1a r c s in 4. dxxx x? ? )1(arctan 22 解 . ??? ???? dtttdttttttxdxxx x )1( c s cs e cs e ct a nt a n)1(a r c t a n 222222 令 11 222 21c o tc o t21c o tc s c tdtttttdtdttdttt ????????? ??? ? cxxxx xctttt ??????????? 222 )( a r c t a n21|1|lna r c t a n21|s i n|lnc o t cxxxx x ?????? 222 )( ar c t an211ln21ar c t an 十三 . 求下列不定積分 : 1. ? ? dxxx 23 4 解 . ??? ??? dtttdtttttxdxxx 23323 c o ss i n32c o s2c o s2s i n8s i n24 令 ctttddttt ?????? ? 5322 c o s532c o s332c o sc o s)c o s1(32 cxx ?????? 252232 )4(51)4(34 2. ? ?x ax 22 解 . ? ?? ???? dtt tadtttata tataxx ax2222c o sc o s1t a ns e cs e ct a ns e c令 cxaaaxcatta ??????? a r c c o st a n 22 3. dxeee xxx??? 21 )1( 解 . ud uu uutdttttdtttttedxe ee xxxx c osc oss i n1s i n111 )1(1 )1( 222 ? ? ?? ?????????? 令令 ceecuu xx ??????? 21a r c s inc o s 4. ?? dxxaxx 2 (a 0) 解 . ?? dxxaxx 2 xu?令 ?? duuau 2422 tau sin2?令 ? tdta 42 sin8
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