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# c - Ch.3 PHYS2041 Problems in Quantitative Methods for...

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Ch.3 PHYS2041 Problems in Quantitative Methods for Basic Physics (1 st Term 10/11) 1 Chapter 3 Techniques of Integration 3.1 Method of Substitution One may make the integration of a function simpler by using substitution of variable. To illustrate the method, consider sin cos n ax axdx , with 0 a . ( 3 . 1 . 1 ) If we let sin ua x , then  cos cos du axd ax a axdx  , ( 3 . 1 . 2 ) and 11 sin cos sin cos n nn ax axdx ax a axdx u du aa  . (3.1.3) The integral of n u is simple and the result is 1 ,1 , 1 ln , 1; n n u Cn udu n uC n   ( 3 . 1 . 4 ) By substituting sin x , one yields: 1 sin , 1 sin cos ln sin . n n ax na ax axdx ax a   ( 3 . 1 . 5 ) Another example is to evaluate 22 dx ax , with . In this case, we try the substitution: sin x at , cos dx a tdt , ( 3 . 1 . 6 ) 2 2 1s i n c o s axa ta t  . ( 3 . 1 . 7 )

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Ch.3 PHYS2041 Problems in Quantitative Methods for Basic Physics (1 st Term 10/11) 2 The integral then becomes 22 cos cos cos cos dx a tdt a tdt dt t C at ax a t   , (3.1.8) where depends on sign of cos t . By substituting 1 sin x t a    , 1 sin dx x C a  . ( 3 . 1 . 9 ) In particular, if we choose t between - /2 and /2, cos 0 t and the ambiguous sign of the integral is “+”. Some useful substitutions for integration: Form Substitution sin x or cos x tan x or sinh x x a sec x or cosh x When the method of substitution is used to evaluate the definite integrals, the lower
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c - Ch.3 PHYS2041 Problems in Quantitative Methods for...

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