1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

Appendix: Mathematical References 437

b.

∫a

a

f(x)dx= 0

c.

∫b

a

f(x)dx=−

∫a

b

f(x)dx

d.

∫b

a

f(x)dx=

∫c

a

f(x)dx+

∫b

c

f(x)dx

3.Derivatives of integrals

a.

d

dt

∫b

a

f(x,t)dx=

∫b

a

∂f

∂t(x,t)dx

b. dtd

∫t

a

f(x)dx=f(t) (Fundamental theorem of calculus;

ais constant)

c. d

dt

∫v(t)

u(t)

f(x,t)dx=f

(

v(t),t

)

v′(t)−f

(

u(t),t

)

u′(t)

+

∫v(t)

u(t)

∂f

∂t(x,t)dx (Leibniz’s rule)

4.Integration by parts

a.

∫

uv′dx=uv−

∫

vu′dx

b.

∫

uv′′dx=v′u−vu′+

∫

vu′′dx

5.Functions defined by integrals

a.Natural logarithm

ln(x)=

∫x

1

dz

z

b.Sine-integral function

Si(x)=

∫x

0

sin(z)

z

dz

c.Normal probability distribution function

(x)=

1

√

2 π

∫x

−∞

e−z^2 /^2 dz

d.Error function

erf(x)=

2

√π

∫x

0

e−z^2 dz

Note: erf(x)= 2 

(√

2 x

)

− 1