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

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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 )
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
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
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online