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GENERAL METHODS OF INTEGRATION 
35 
14. Integration by Parts. The method is contained in the formula 
(1) J udv = uo — J*vdu; 
Introduction to the Calculus, p. 243. The cases to which the method 
applies form a restricted class, but one which, in an extended study 
of integration, must be recognized. 
The method is best studied through typical examples like those 
of the paragraph to which reference has been made, and the student 
should now review that paragraph. There is little point in multi 
plying examples here, since such examples would carry with them 
the direction to use this method, and the whole difficulty lies in the 
fact that, in practice, the student is not told when to use the method. 
For this reason he should strive to detect those integrals in the mis 
cellaneous list at the end of the chapter (cf. also § 10) which are best 
evaluated in this manner.* Perhaps a single example maybe useful. 
Example. To evaluate the integral 
J =/ 
, a 2 + x 2 , 
a; tan 1 —— ! —dx. 
u= tan- 1 ^ 
the object being to eliminate the transcendental function through 
differentiation. Then 
du = 
and we have by (1) : 
T x 2 . . a 2 + x 1 , /* x 3 < 
I = — tan 1 — a 2 I 
2 a 2 J a 4 + (a : 
To evaluate the latter integral, let y = x 2 :