tabela integrais

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<p>16 INDEFINITE INTEGRALS</p><p>Definition of an Indefinite Integral</p><p>If dy</p><p>dx</p><p>f x= ( ), then y is the function whose derivative is f(x) and is called the anti-derivative of f(x) or the indefi-</p><p>nite integral of f(x), denoted by f x dx( ) .∫ Similarly if y f u du= ∫ ( ) , then dy</p><p>du</p><p>f u= ( ). Since the derivative of a</p><p>constant is zero, all indefinite integrals differ by an arbitrary constant.</p><p>For the definition of a definite integral, see 18.1. The process of finding an integral is called integration.</p><p>General Rules of Integration</p><p>In the following, u, � , w are functions of x; a, b, p, q, n any constants, restricted if indicated; e = 2.71828 … is</p><p>the natural base of logarithms; ln u denotes the natural logarithm of u where it is assumed that u > 0 (in general,</p><p>to extend formulas to cases where u < 0 as well, replace ln u by ln |u|); all angles are in radians; all constants of</p><p>integration are omitted but implied.</p><p>16.1. a dx ax=∫</p><p>16.2. af x dx a f x dx( ) ( )= ∫∫</p><p>16.3. ( )u w dx u dx dx w dx± ± ± = ± ± ±∫∫ ∫ ∫� �� �</p><p>16.4. u d u du� � �= −∫ ∫ (Integration by parts)</p><p>For generalized integration by parts, see 16.48.</p><p>16.5. f ax dx</p><p>a</p><p>f u du( ) ( )=∫ ∫</p><p>1</p><p>16.6. F f x dx F u</p><p>dx</p><p>du</p><p>du</p><p>F u</p><p>f x</p><p>du u{ ( )} ( )</p><p>( )</p><p>( )</p><p>= = ′ =∫ ∫∫ where ff x( )</p><p>16.7. u du</p><p>u</p><p>n</p><p>n nn</p><p>n</p><p>=</p><p>+</p><p>≠ − = −</p><p>+</p><p>∫</p><p>1</p><p>1</p><p>1 1 16 8, ( , . )For see</p><p>16.8.</p><p>du</p><p>u</p><p>u u u u</p><p>u</p><p>= > − <</p><p>=</p><p>∫ ln ln( )</p><p>ln | |</p><p>if or if0 0</p><p>16.9. e du eu u=∫</p><p>16.10. a du e du</p><p>e</p><p>a</p><p>a</p><p>a</p><p>a au u a</p><p>u a u</p><p>= = = > ≠∫ ∫ ln</p><p>ln</p><p>ln ln</p><p>, ,0 1</p><p>67</p><p>16.11. sin cosu du u∫ = −</p><p>16.12. cos sinu du u∫ =</p><p>16.13. tan ln sec ln cosu du u u∫ = = −</p><p>16.14. cot ln sinu du u∫ =</p><p>16.15. sec ln (sec tan ) ln tanu du u u</p><p>u∫ = + = +⎛</p><p>⎝⎜</p><p>⎞</p><p>⎠⎟2 4</p><p>π</p><p>16.16. csc ln(csc cot ) ln tanu du u u</p><p>u∫ = − =</p><p>2</p><p>16.17. sec tan2 u du u∫ =</p><p>16.18. csc cot2 u du u∫ = −</p><p>16.19. tan tan2 u du u u∫ = −</p><p>16.20. cot cot2 u du u u∫ = − −</p><p>16.21. sin</p><p>sin</p><p>( sin cos )2</p><p>2</p><p>2</p><p>4</p><p>1</p><p>2</p><p>u du</p><p>u u</p><p>u u u∫ = − = −</p><p>16.22. cos</p><p>sin</p><p>( sin cos )2</p><p>2</p><p>2</p><p>4</p><p>1</p><p>2</p><p>u du</p><p>u u</p><p>u u u∫ = + = +</p><p>16.23. sec tan secu u du u∫ =</p><p>16.24. csc cot cscu u du u∫ = −</p><p>16.25. sinh coshu du u=∫</p><p>16.26. cosh sinhu du u=∫</p><p>16.27. tanh ln coshu du u=∫</p><p>16.28. coth ln sinhu du u=∫</p><p>16.29. sech oru du u eu= − −∫ sin (tanh ) tan1 12</p><p>16.30. csch oru du</p><p>u</p><p>eu∫ = − −ln tanh coth</p><p>2</p><p>1</p><p>16.31. sech2u du u∫ = tanh</p><p>INDEFINITE INTEGRALS68</p><p>16.32. csch2∫ = −u du ucoth</p><p>16.33. tanh tanh2∫ = −u du u u</p><p>16.34. coth coth2∫ = −u du u u</p><p>16.35. sinh</p><p>sinh</p><p>(sinh cosh )2 2</p><p>4 2</p><p>1</p><p>2∫ = − = −u du</p><p>u u</p><p>u u u</p><p>16.36. cosh</p><p>sinh</p><p>(sinh cosh )2 2</p><p>4 2</p><p>1</p><p>2∫ = + = +u du</p><p>u u</p><p>u u u</p><p>16.37. sech sechu u du utanh = −∫</p><p>16.38. csch cschu u du ucoth = −∫</p><p>16.39.</p><p>du</p><p>u a a</p><p>u</p><p>a2 2</p><p>11</p><p>+ = −∫ tan</p><p>16.40.</p><p>du</p><p>u a a</p><p>u a</p><p>u a a</p><p>u</p><p>a</p><p>u a2 2</p><p>1 2 21</p><p>2</p><p>1</p><p>− = −</p><p>+</p><p>⎛</p><p>⎝⎜</p><p>⎞</p><p>⎠⎟ = − >∫ −ln coth</p><p>16.41.</p><p>du</p><p>a u a</p><p>a u</p><p>a u a</p><p>u</p><p>a</p><p>u a2 2</p><p>1 2 21</p><p>2</p><p>1</p><p>− = +</p><p>−</p><p>⎛</p><p>⎝⎜</p><p>⎞</p><p>⎠⎟ = <∫ −ln tanh</p><p>16.42.</p><p>du</p><p>a u</p><p>u</p><p>a2 2</p><p>1</p><p>−</p><p>=∫ −sin</p><p>16.43.</p><p>du</p><p>u a</p><p>u u a</p><p>u</p><p>a2 2</p><p>2 2 1</p><p>+</p><p>= + + −∫ ln( ) sinhor</p><p>16.44.</p><p>du</p><p>u a</p><p>u u a</p><p>2 2</p><p>2 2</p><p>−</p><p>= + −∫ ln ( )</p><p>16.45.</p><p>du</p><p>u u a a</p><p>u</p><p>a2 2</p><p>11</p><p>−</p><p>= −∫ sec</p><p>16.46.</p><p>du</p><p>u u a a</p><p>a u a</p><p>u2 2</p><p>2 21</p><p>+</p><p>= − + +⎛</p><p>⎝⎜</p><p>⎞</p><p>⎠⎟∫ ln</p><p>16.47.</p><p>du</p><p>u a u a</p><p>a a u</p><p>u2 2</p><p>2 21</p><p>−</p><p>= − + −⎛</p><p>⎝⎜</p><p>⎞</p><p>⎠⎟∫ ln</p><p>16.48. f g dx f g f g f g fn n n n n( ) ( ) ( ) ( ) ( )= − ′ + ′′ − −− − −∫ 1 2 3 1� gg dxn( )∫</p><p>This is called generalized integration by parts.</p><p>INDEFINITE INTEGRALS 69</p><p>Important Transformations</p><p>Often in practice an integral can be simplified by using an appropriate transformation or substitution together</p><p>with Formula 16.6. The following list gives some transformations and their effects.</p><p>16.49. F ax b dx</p><p>a</p><p>F u du( ) ( )+ = ∫∫</p><p>1</p><p>where u = ax + b</p><p>16.50. F ax b dx</p><p>a</p><p>u F u du( ) ( )+ =∫ ∫</p><p>2</p><p>where u ax b= +</p><p>16.51. F ax b dx</p><p>n</p><p>a</p><p>u F u dun n( ) ( )+ =∫ ∫ −1 where u ax bn= +</p><p>16.52. F a x dx a F a u u du( ) ( cos ) cos2 2− = ∫∫ where x = a sin u</p><p>16.53. F x a dx a F a u u du( ) ( sec )sec2 2 2+ = ∫∫ where x = a tan u</p><p>16.54. F x a dx a F a u u u du( ) ( tan )sec tan2 2− =∫ ∫ where x = a sec u</p><p>16.55. F e dx</p><p>a</p><p>F u</p><p>u</p><p>duax( )</p><p>( )= ∫∫</p><p>1</p><p>where u = eax</p><p>16.56. F x dx F u e duu(ln ) ( )= ∫∫ where u = ln x</p><p>16.57. F</p><p>x</p><p>a</p><p>dx a F u u dusin ( ) cos−⎛</p><p>⎝⎜</p><p>⎞</p><p>⎠⎟</p><p>=∫ ∫1 where u</p><p>x</p><p>a</p><p>= −sin 1</p><p>Similar results apply for other inverse trigonometric functions.</p><p>16.58. F x x dx F</p><p>u</p><p>u</p><p>u</p><p>u</p><p>du</p><p>(sin , cos ) ,= +</p><p>−</p><p>+</p><p>⎛</p><p>⎝⎜</p><p>⎞</p><p>⎠⎟∫ ∫2</p><p>2</p><p>1</p><p>1</p><p>1 12</p><p>2</p><p>2 ++ u2 where u</p><p>x= tan</p><p>2</p><p>INDEFINITE INTEGRALS70</p>