
“real: chapter_08” — 2011/5/22 — 23:34 — page 34 — #34
8-34 Real Analysis
d
c
f (x)dx = F(d) = F(φ(β)) = G(β) + k
= G (β) − G(α)
=
β
a
f (φ(t))φ
(t)dt −
α
a
f (φ(t))φ
(t)dt
=
β
α
f (φ(t))φ
(t)dt.
Here α may be less than β or equal to β or greater than β and
β
α
f (φ(t))φ
(t)dt is as defined in Remark 8.5.10.
This theorem on change of variable is very efficient in the
computation of integrals as seen by the following examples.
Examples 8.6.4
1. Consider
π
0
x sin xdx. Let φ : [0, π]→[0, π] be defined by
φ(t) = π − t so that φ is differentiable and (φ
(t) =−1)
is continuous. Further φ([0, π]) =[0, π] and φ(0) = π,
φ(π) = 0. We now apply Theorem 8.6.3 to get
π
0
x sin ...