The Chemistry Maths Book, Second Edition

6.7 Parametric differentiation of integrals 185

In addition, by integration by parts,

It follows that

so that the order of integration with respect to xand differentiation with respect to α

can be interchanged in this case. This result is true in the general case

(6.34)

iff(x, α)and are continuous functions of xand α:

(6.35)

For the corresponding definite integral, if the limits are independent of the

parameter α,

(6.36)

When one or both of the limits of integration are infinite, it is necessary to ensure that

the integral of the function and that of its derivative are both convergent.

EXAMPLE 6.22Integrate.

The integral was evaluated in Section 6.5 from a reduction formula derived by

successive integrations by parts. An alternative method is to differentiate the simple

standard integral

The nth derivative ofe

−ax

with respect to ais(−1)

n

1 x

n

1 e

−ax

so that

ZZ

00

11

1

∞∞

xe dx

d

da

edx

d

da

nax n

n

n

ax n

n

n

−−

=−() =−()

aa

n

a

n













=

!

+ 1

Z

0

1

0

∞

edx

a

a

−ax

=, ()≠

Z

0

∞

exdx

−ax n

d

d

fx dx

d

d

fx dx

d

d

Fb

a

b

a

b

α

α

α

α

α

ZZ(),= (), (











 =,αα

α

)()−,α

d

d

Fa

d

d

fx dx

d

d

fx dx

d

d

Fx

α

α

α

α

α

ZZ(),= (), ()α













=,

d

d

fx

α

(),α

Zfx dx Fx C() (),=,+αα

d

d

edx

d

d

edx

xx

αα

αα

ZZ

−−





















=

ZZ

d

d

edx xedx

x

xx

α

α

α

−αα





















−

=− = +













1

2

ee

−αx