Definite Integrals, with Physical Applications. 365 



(3) Means of facilitating the Calculus off(t). 



7. By this method we may always revert from <f> (x) to f(t) ; 

 but to facilitate the process, it will be found convenient to reduce 

 (p(.v) to its simplest form, previous to the application of the 

 general theorem. 



Thus suppose <p(x) expanded according to the descending powers 

 of x + m (m being any positive quantity) i. e. if 



* {X) = FhS + (J+iji + {x^tmf +&C - 

 then assuming the result of the second Example above, we get, 



/(0 = r-' {A^A^l. (i) + \ 2 + l 2 J J +&c .|. 



8. When <J>(x) is a rational fraction, the highest power of x 

 in the denominator being greater than that in the numerator, 

 and the denominator not vanishing in the interval from x = o 

 to oo», then if we decompose <p(x) into its simple fractions, the 

 results of the first and second Examples above will give us the 

 value of f(t). 



Ex. 4. Given 



A to- *-(*-l).(*-fl) (x-n + 1) 



(py > (* + l)(* + 2)(* + S) (* + » + l)' 



to find/(0- Put 



X + l # + 2 .T + W + l' 



which compared with the above, gives 



x.(x-l).(x-2)...(x-n + 1) = Ni (x + 2) (x + 3)...(x + n + 1) 

 + A r , (x + 1) (x + 3)...(x + « + !) + &c. 



3 a2 



