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Mathematics
Problem
Let
a
a
a
be a positive constant. Evaluate the improper integral
L
(
a
)
=
∫
0
∞
e
−
a
x
d
x
L(a)=\displaystyle \int_{0}^{\infty} e^{-ax}\,dx
L
(
a
)
=
∫
0
∞
e
−
a
x
d
x
. (a) Verify directly from the definition of an improper integral that
L
(
a
)
L(a)
L
(
a
)
converges for all
a
>
0
a>0
a
>
0
. (b) Calculate
L
(
a
)
L(a)
L
(
a
)
as an explicit function of
a
a
a
.
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