164 Chapter 6Methods of integration
(6.3)
(6.4)
(6.5)
(6.6)
More generally, the relations can be used to express a functionsin
m1 x 1 cos
n1 x, where
mand nare positive integers, in terms of simple sines and cosines, but alternative
methods of integration are often simpler to use when mor nis greater than 2.
Table 6.1
For a 1 ≠ 1 b:
EXAMPLES 6.1The integrals in Table 6.1
Integral 1:
By equation (6.1),cos cos. Therefore
22
1
2
xx=+ 14
Zcos
22 xdx
Zsin cos
cos( ) cos( )
ax bx dx
abx
ab
abx
ab
=−
−
−
1
2
+C
Zcos cos
sin( ) sin( )
ax bx dx
abx
ab
abx
ab
=
−
−
1
2
+C
Zsin sin
sin( ) sin( )
ax bx dx
abx
ab
abx
ab
=
−
−
−
1
2
+C
Zsin cosax ax dx sin
a
=+ax C
1
2
2
Zsin sin cos
2
1
2
ax dx
a
=−ax ax ax C
Zcos sin cos
2
1
2
ax dx
a
=+ax ax ax C
sin cosx y=−++sin(xy xy) sin( )
1
2
cos cosxy=−++cos(xy xy) cos( )
1
2
sin sinx y=−−+cos(xy) cos(xy)
1
2
sin cosxx x= sin
1
2
2