6.3 The method of substitution 167
and, substituting in (6.7),
(6.9)
The essential skill in applying the method of substitution is the ability to recognize
when an integrand can be written in the form (6.8). The transformation of the integral
is then achieved by the substitution
u 1 = 1 u(x), (6.10)
or, alternatively,x 1 = 1 x(u),
EXAMPLE 6.2Show that
where the numbers a,b, and nare arbitrary, except thatn 1 ≠ 1 − 1.
Letu 1 = 1 ax 1 + 1 b. Thendu 1 = 1 a 1 dx,and
0 Exercises 11, 12
EXAMPLE 6.3Integrate.
Because we make the substitution u 1 = 1 x 1 + 1 x
2. Then
du 1 = 1 (1 1 + 12 x) 1 dxand
Therefore
The result is confirmed by differentiation:
0 Exercises 13, 14
d
dx
21 xx C xx 2 x
212 2 12() ()()++
=+ +
−ZZ()()xx++= =+=+xdx u du u C xx()
212 −− 12 12 21212 2 2 ++C
fx x x x u
du
dx
() ( ) ( )=+ + =
212 −− 1212
()(),12
2+= +x
d
dx
xx
ZZfxdx x x() =+( ) ( )+xdx
212 −12
ZZ()
()
(
ax b dx
a
udu
a
u
n
C
ax b
an
nnnn+= =
+=
++11
1
11++
C
Z()
()
ax b dx ()
an
ax b C
nn+=
++
+1
1
1dx
dx
du
= du.
du
du
dx
= dx
ZZ Zfxdx gu
du
dx
() ==() dx g u du()