Mathematical Methods for Physics and Engineering : A Comprehensive Guide

5.13 EXERCISES

constant limits of integration the order of integration and differentiation can be

reversed.

In the more general case where the limits of the integral are themselves functions

ofx, it follows immediately that

I(x)=

∫t=v(x)

t=u(x)

f(x, t)dt

=F(x, v(x))−F(x, u(x)),

which yields the partial derivatives

∂I

∂v

=f(x, v(x)),

∂I

∂u

=−f(x, u(x)).

Consequently

dI

dx

=

(

∂I

∂v

)

dv

dx

+

(

∂I

∂u

)

du

dx

+

∂I

∂x

=f(x, v(x))

dv

dx

−f(x, u(x))

du

dx

+

∂

∂x

∫v(x)

u(x)

f(x, t)dt

=f(x, v(x))

dv

dx

−f(x, u(x))

du

dx

+

∫v(x)

u(x)

∂f(x, t)

∂x

dt, (5.47)

where the partial derivative with respect toxin the last term has been taken

inside the integral sign using (5.46). This procedure is valid becauseu(x)andv(x)

are being held constant in this term.

Find the derivative with respect toxof the integral

I(x)=

∫x 2

x

sinxt

t

dt.

Applying (5.47), we see that

dI

dx

=

sinx^3

x^2

(2x)−

sinx^2

x

(1) +

∫x 2

x

tcosxt

t

dt

=

2sinx^3

x

−

sinx^2

x

+

[

sinxt

x

]x 2

x

=3

sinx^3

x

− 2

sinx^2

x

=

1

x

(3 sinx^3 −2sinx^2 ).

5.13 Exercises

5.1 Using the appropriate properties of ordinary derivatives, perform the following.