6.3 The method of substitution 165
Integral 3:
By equation (6.3), Therefore
where C′is a new arbitrary constant. Most tabulations of indefinite integrals omit the
arbitrary constant, and this example shows why different tabulations sometimes give
apparently different values for indefinite integrals.
Integral 4:
By equation (6.4),
Therefore
0 Exercise 1–10
6.3 The method of substitution
The polynomial
f(x) 1 = 1 (2x 1 − 1 1)
3can be integrated by first expanding the cube and then integrating term by term (see
the corresponding discussion of the chain rule in Section 4.6):
ZZsin sin cos cos
sin
24
1
2
26
1
2
2
x x dx=−x x dx
=
xxx
C
2
6
6
−
sin
sin sin cos( ) cos cos c 24
1
2
26
1
2
xx=−−x x 2 x
=−oos. 6x
Zsin 2 sin 4xxdx
=+′
1
4
2
2sin xC
=− −
+= +−
1
8
12 2
1
4
2
1
8
22sin xCsin xC
ZZsin cos sin cos 22
1
2
4
1
8
xxdx==xdx−+ 4 xC
sin cos sin. 22
1
2
xx x= 4
Zsin 2 cos 2xxdx
=+
xxxC+
1
4
222sin cos
ZZcos cos sin
22
1
2
14
1
2
1
4
xdx=+x dx x 4 x
=+
+C