082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

14 Techniques
ofintegration
Contents 14.1 Introduction 457
14.2 Integrationbyparts 457
14.3 Integrationbysubstitution 463
14.4 Integrationusingpartialfractions 466
Reviewexercises14 468
14.1 INTRODUCTION
The previous chapter showed us how to integrate functions which matched the list of
standard integrals given in Table 13.1. Clearly, it is impossible to list all possible functions
in the table and so some general techniques are required. Integration techniques
maybeclassifiedasanalytical,thatisexact,ornumerical,thatisapproximate.Wewill
now study threeanalytical techniques:
(1) integration by parts;
(2) integration by substitution;
(3) integration usingpartialfractions.
14.2 INTEGRATIONBYPARTS
This technique is used to integrate a product, and is derived from the product rule for
differentiation. Let u and v be functions of x. Then the product rule of differentiation
states:
d du
(uv) =
dx dx v +udv dx
Rearranging we have
u dv
dx = d dx (uv)−vdu dx