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

430 Chapter 13 Integration
Example13.2 Find d dx
( x
n+1
n +1 +c )
and hence deduce that ∫ x n dx= xn+1
n +1 +c.
Solution From Table 10.1 wefind
( )
d x
n+1
dx n +1 +c = d ( ) x
n+1
+ d dx n +1 dx (c) usingthe linearity
of differentiation
= 1 d
n +1dx (xn+1 ) + d dx (c)
= 1
n +1 {(n +1)xn } +0
=x n
again usingthe
linearityof differentiation
usingTable10.1
Consequently, reversing the process wefind
∫
x n dx= xn+1
n +1 +c
as required. Note that this result is invalid if n = −1 and so this result could not be
applied tothe integral ∫ (1/x)dx.
Table 13.1 lists several common functions and their integrals. Although the variable
x is used throughout Table 13.1, we can use this table to integrate functions of other
variables, forexamplet andz.
Example13.3 UseTable 13.1 tointegratethe following functions:
(a) x 4
(b) coskx, wherekisaconstant
(c) sin(3x +2)
(d) 5.9
(e) tan(6t −4)
(f) e −3z
1
(g)
x 2
(h) cos100nπt, wherenisaconstant
Solution (a) From Table 13.1, we find ∫ x n dx= xn+1
n +1 +c,n ≠ −1. To find ∫ x 4 dxletn=4;
weobtain
∫
x 4 dx= x5
5 +c
(b) FromTable 13.1, wefind ∫ cos(ax)dx = sin(ax) +c. Inthiscasea =kand so
a
∫
coskxdx = sinkx +c
k