XXvi USEFUL FORMULAS. 



b. Definite Integration. 



n 



be n 



j </) {x) dx ^=.\ ^ ipc) dx -\- \ ^ ipc) dx A^ . . . I ^ {x) dx. 

 a a b m 



b a 



i <l> {x) dx ^ — \ 4> (x) dx. 

 a b 



a a 



\ (\> {x) dx'=\<^ {a — X) dx. 

 o o 



If <ji (x) = <j> (— x), an " even function " of x, 



a o a. 



j c^ {x) dxz=i\<^ ix) dx=:.\\<^ ipc) dx. 

 o — a — a 



li (f) {x) = — <f> (— x), an " odd function " of x, 



o a ~\~ ^ 



j </) (x) dx = i 4> (x) dx, and | </> (x) dx = o. 

 —a o — a 



If A be the greatest and B the least value of (j) (x) within the limits a and b, 



b 

 A {b — a)>C<f> (x) dx > B (b — a), 

 a 



a formula useful in determining approximate values of integrals. See, e. g., 

 section 6, d. 

 b 



U u = i (t> (x) dx, 

 a 



CO 



/dx , 



o 



CO 



+ 



CO 



C dx _ r dx _ 



J I +;c^~"J I J^x^~^''' 

 o I 



00 ^ ^ 



o o 



