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Lebl-Basic-Analysis-I
· Problem 5.3.5
The Riemann Integral
Mathematics · integration-by-parts
Problem
Prove integration by parts. That is, suppose
F
F
F
and
G
G
G
are continuously differentiable functions on
[
a
,
b
]
[a,b]
[
a
,
b
]
. Then prove
∫
a
b
F
(
x
)
G
′
(
x
)
d
x
=
F
(
b
)
G
(
b
)
−
F
(
a
)
G
(
a
)
−
∫
a
b
F
′
(
x
)
G
(
x
)
d
x
.
\int_a^b F(x) G'(x)\,dx = F(b)G(b) - F(a)G(a) - \int_a^b F'(x) G(x)\,dx.
∫
a
b
F
(
x
)
G
′
(
x
)
d
x
=
F
(
b
)
G
(
b
)
−
F
(
a
)
G
(
a
)
−
∫
a
b
F
′
(
x
)
G
(
x
)
d
x
.
Related theory
integration-by-parts
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