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Tabelas de Integrais Indefinidas Observação: Em todas as fórmulas, a constante arbitrária é omitida; α,,, cba representam números reais e qpnm ,,, inteiros positivos. Quando 2a aparece no integrando, a deve ser tomado como um número positivo, ln( ) pode sempre ser substituído por ln | | . 1. ∫ = cxcdx 2. ∫ ∫= dxxfcdxxcf )()( 3. ( ) ∫∫ ∫ +=+ dxxgdxxfdxxgxf )()()()( 4. ∫ ∫−= dxxfxgxgxfdxxgxf )()()()()()( '' 5. ∫ ∫−= vduuvudv 6. ∫ + = + 1 1 a xdxx a a , 1≠a 7. ∫ = ||ln1 xdxx 8. ∫ = |)(|ln)( )(' xfdx xf xf 9. ∫ = a edxe ax ax 10. ∫ = )ln(a adxa x x 11. ∫ −= xxxdxx )ln()ln( 12. [ ] 1;)ln()(log)ln()ln( 1)(log ≠−=−=∫ xa x xxxxx a dxx aa 13. ∫ −= = + −− a xCotg aa x tg aax dx 11 22 11 14. ∫ + − = −= − − ax ax aa x tgh aax dx ln 2 11 1 22 15. ∫ <> −− +− − > = + − 0;0;ln 2 1 0;1 1 2 ba abx abx ab ab a abx tg ab bxa dx 16. ∫ + = + b bxa bbxa xdx 2 2 ln2 1 17. 1;)()1(2 32 )( 1 )1(2 1 )( 12122 >+×− − + + × − = + ∫∫ −− m bxa dx am m bxamabxa dx mmm 18. 1;))((1(2 1 )( 122 >+− − = +∫ − m bxambbxa xdx mm 19. −= = − −− ∫ a x a x sen xa dx 11 22 cos 20. )ln( 22 22 axx ax dx ±+= ± ∫ 21. ∫ = − − x a aaxx dx 1 22 cos 1 22. ∫ ++ −= ± x xaa axax dx 22 22 ln1 23. ( )( )∫ ±+±±=± 2222222 ln21 axxaaxxdxax 24. ∫ ++ −+= + x axa aaxdx x ax 2222 22 ln 25. ∫ −−= − − x a aaxdx x ax 122 22 cos 26. ∫ ±= ± 22 22 ax ax xdx 27. ( ) 2/32222 3 1 ∫ ±=± axdxaxx 28. ( ) ( ) ( )[ ]∫ ±++±±±=± 2242222/3222/322 ln33281 axxaaxxaaxxdxax 29. ( )∫ ± ± = ± 222 2/322 axa x ax dx 30. ( ) ( ) ( ) ( )( )2242222/322222 ln 884 axx a ax xa ax xdxaxx ±+−±±=±∫ m 31. ∫ +−=− − a x senaxaxdxxa 122222 2 1 32. ∫ −+ −−= − x xaa axadx x xa 2222 22 ln 33. ∫ −+ −= − x xaa a dx xax 22 22 ln11 34. ∫ −−= − 22 22 xadx xa x 35. ∫ ∫ ××=+ duuuautgafadxaxxf )(sec))sec(),((),( 222 ; )(utgax ×= 36. ∫ ∫ ×××=− duutguutgauafadxaxxf )()sec())(),sec((),( 22 ; )sec(uax ×= 37. ∫ ∫ ××−=− duusenusenauafadxxaxf )())(),cos((),( 22 ; )cos(uax ×= 38. ∫ ∫ +− = dy a dy a byf a dxXxf 2 , 1),( abyx /)( −= ; 2bacd −= ; cbxaxX ++= 22 39. ( ) ( )∫∫ − + + ++ + ++ + =+ dxbxax nm an nm bxaxdxbxax nn nn nn 1 1 11 )( 40. )ln(1 bxa bbxa dx += +∫ 41. [ ])ln(12 bxaabxabbxa xdx +−+= +∫ 42. + ++= +∫ bxa abxa bbxa xdx )ln(1)( 22 43. +− + +− − = + −−∫ 122 ))(1())(2( 11 )( mmm bxam a bxambbxa xdx ; 3≥m 44. ∫ +=+ 2)( 3 2 bxa b dxbxa 45. [ ]∫ ∫ +−++=+ − dxbxaxmabxaxbmdxbxax mmm 12)()32( 2 46. ∫∫ +− − − − +− = + −− )()22( )32( )1( )( )( 11 bxax dx am bm xma bxa bxax dx mmm ; 1≠m 47. ∫ ∫ − =+ zdzz b azf b dxbxaxf ,2),( 2 ; bxaz +=2 48. − + +− + = + − ∫ 3 23)(ln 2 1 3 1 1 22 2 222 a ax tg xaxa xa axa dx 49. )cos()( xdxxsen −=∫ 50. )()cos( xsendxx =∫ 51. ))ln(cos()( xdxxtg −=∫ 52. ))(ln()(cot xsendxxg =∫ 53. +=∫ 22 ln)sec( pixtgdxx 54. =∫ 2 ln)(cos xtgdxxec 55. [ ])()cos( 2 1)(2 xsenxxdxxsen −=∫ 56. ∫∫ − − − + − = dxxsen m m m xsenxdxxsen m m m 2 1 )(1)()]cos()( 57. )2( 4 1 2 1)(cos2 xsenxdxx +=∫ 58. dxx m m m xxsendxx m m m ∫∫ − − − += )(cos1)(cos)()(cos 2 1 59. )()(sec)(cos 2 2 xtgdxxx dx ∫∫ == 60. ∫∫ −− − − + − = )(cos1 2 )(cos)1( )( )(cos 21 x dx m m xm xsen x dx mmm ; 1>m 61. )(cot)(cos)( 2 2 xgdxxecxsen dx ∫∫ −== 62. ∫∫ −− − − + − − = )(1 2 )()1( )cos( )( 21 xsen dx m m xsenm x xsen dx mmm ; 1>m 63. ∫ = ± 24)(1 x tg xsen dx mm pi 64. ∫ = + 2)cos(1 x tg x dx 65. ∫ −= − 2 cot)cos(1 xg x dx 66. ∫ > − + − > −++ −−+ − = + − 22 22 1 22 22 22 22 22 ;22 ; 2 2ln1 )( ba ba bxtga tg ba ab abbxtga abbxtga ab xsenba dx 67. ∫ > + − − > −− − ++ − − = + − 22 22 1 22 22 22 22 22 ; 22 ; 2 2ln1 )cos( ba ba x tgba tg ba ab baxtgab baxtgab ab xba dx 68. 22;)(2 )( )(2 )()()( nm nm xnmsen nm xnmsendxmxsennxsen ≠ + + − − − =×∫ 69. 22;)(2 )cos( )(2 )cos()cos()( nm nm xnm nm xnmdxmxnxsen ≠ + + − − − =×∫ 70. 22;)(2 )( )(2 )()cos()cos( nm nm xnmsen nm xnmsendxmxnx ≠ + + + − − =×∫ 71. 1;)( 1 )()( 2 1 ≠− − = ∫∫ − − ndxxtg n xtgdxxtg n n n 72. ∫ = ))(ln()cos()( xtgxxsen dx 73. ∫ ∫ >+ − = −− 1;)(cos)()(cos)1( 1 )(cos)( 11 mxxsen dx xmxxsen dx mmm 74. ∫∫ −+−= dxxxmxxdxxsenx mmm )cos()cos()( 1 75. ∫∫ − −= dxxsenxmxsenxdxxx mmm )()()cos( 1 76. 211 1)()( xxsenxdxxsen −+=∫ −− 77. 211 1)(cos)(cos xxxdxx −−=∫ −− 78. ( )211 1ln 2 1)()( xxtgxdxxtg +−=∫ −− 79. ( )211 1ln 2 1)(cot)(cot xxgxdxxg ++=∫ −− 80. ( ) ( ) )(122)()( 122121 xsenxxxsenxdxxsen −−− −+−=∫ 81. ( ) ( ) )(cos122)(cos)(cos 122121 xxxxxdxx −−− −−−=∫ 82. dx x x nn xsenxdxxsenx nn n ∫∫ − + − + = +−+ − 2 111 1 11 1 1 )()(83. dx x x nn xxdxxx nn n ∫∫ − + + + = +−+ − 2 111 1 11 1 1 )(cos)(cos 84. 4 )ln( 2 )ln( 22 x x xdxxx −=∫ 85. 1;)1(1)ln( 2 121 ≠ + − + = ++ ∫ mm x m xdxxx mMn m 86. ( ) ( ) ( )∫∫ −−= dxxqxxdxx qqq 1)ln()ln()ln( 87. ( ) ( )∫ += + 1 )ln()ln( 1 q xdx x x qq 88. ∫ = ))ln(ln()ln( xdxxx dx 89. ∫∫ ≠+ − + = − + 1;))(ln( 11 ))(ln())(ln( 1 1 mdxqxx m q m xxdxxx m qm qm 90. ∫ −= ))cos(ln(2 1))(ln( 2 1))(ln( xxxsenxdxxsen 91. ∫ += ))cos(ln(2 1))(ln( 2 1))cos(ln( xxxsenxdxx 92. )1(2 −=∫ axa edxex ax ax 93. ∫∫ − −= dxex a exdxex axm axm axm 1 ; 0>m 94. 1; 1)1( 11 >−+−−= ∫∫ −− mdxx e m a xm edx x e m ax m ax m ax 95. ∫∫ −= dxx e aa xedxxe axax ax 1)ln()ln( 96. ∫ + − = 22 ))cos()(()( na nxnnxsenaedxnxsene ax ax 97. ∫ + + = 22 ))()cos(()cos( nx nxsennnxaedxnxe ax ax 98. ∫ +−=+ )ln(1 ax ax bea aqa x bea dx 99. ∫ = )cosh()( xdxxsenh 100. ∫ = )()cosh( xsenhdxx 101. ∫ = )cosh(ln)( xdxxtgh 102. ∫ = )(ln)(cot xsenhdxxgh 103. ∫ −− == ))((2)(sec 11 xsenhtgetgdxxh x 104. ∫ = 2 ln)(cos xtghdxxech 105. ( )∫ ∫ ∫ = − = + + = )(; 1 2 ; 11 22 ))(( 2 22 xsenu u du uf x tgz z dz z zf dxxsenf 106. ( ) ∫ ∫ ∫ = − − = + + − = )cos(; 1 2 ; 11 12 ))(cos( 2 22 2 xu u du uf x tgz z dz z zf dxxf 107. ( )∫ ∫ ∫ = − − = + + − + = )(; 1 1, 2 ; 11 1 , 1 22 ))cos(),(( 2 2 22 2 2 xsenu u du uuf x tgz z dz z z z zf dxxxsenf 108. = − ××× =× − ××× == ∫∫ KL KL ;7;5;3;1 5 4 3 2 ;6;4;2; 2 1 4 3 2 1 )(cos)( 2/ 0 2/ 0 n n n n n n dxxdxxsen nn pi pipi 109. == +××+×+ −××× == +××+×+ −××× =× +××× −×××−××× =∫ LL L L LL L L K L LL ;7;5;3,;3;2;1;)()3()1( )1(42 ;3;2;1,;7;5;3;)()3()1( )1(42 ;4;2,; 2)(42 )1(31)1(31 )(cos)( 2/ 0 2 nm nmmm n nm mnnn m nm nm nm dxxxsenn pi pi 110. ∫ +∞ ∞− − = pi22/ 2 dxe x
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