184 PRACTICAL MATHEMATICS 



105. (1) To integrate x n log e x. 



Now I x n log e x dx = uv \v du 



Here we can put u = x n or dv = x n dx, but we must differ- 

 entiate x n in the first case and integrate x n in the second case. 

 As we can easily do both operations, it is perfectly immaterial 

 which selection we make. We can also put u = log e x or dv 

 = log e x dx, but we find that although we can easily differentiate 

 log e x, we shall have great difficulty in integrating log e a;, and in 

 consequence we are led to make the selection u = log e x. 



Thus u = log. x and du = - dx 



x 



r x^^" 



dv = x n dx and v = \x n dx = 



J n+l 



r x n ~^ p x ^"^ i 



Then \x n log., x dx = log., x \ dx 



n+ 1 )n+ 1 x 



n + 1 



n+l (n+ I) 2 



n + 



When n = the integral becomes llog e a; dx, and |log e a? das 

 = x(\og e x- 1). 

 (2) To integrate x n ef 



Now \x n eF 1 dx = uv I v du 



If u= x n , then du = nx n ~ l dx 



r tf 1 ^ 



dv = ef 3 * dx, then v = yf dx = - 



Therefore \x n tP* dx = -x n ef* - -fa:"- 1 e dx 

 J a a] 



In this case the result of applying the rule is to produce on 

 the right-hand side an integral of the same form as the original 

 integral, but in which the power of x has been diminished by 1. 

 We can, however, use the above result as a standard form in 

 which n and a can be given their assigned values. 



