1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

438 Appendix: Mathematical References

e.Integrated Bessel function

IJ(x)=

∫x

0

J 0 (z)dz

Table of Integrals

Any letter exceptxrepresents a constant. The integration constants have been
left off.

1.Rational functions

1.1

∫ dx

h+kx=

1

kln|h+kx|

1.2

∫ dx

x^2 +a^2

=^1

a

tan−^1

(

x

a

)

1.3

∫ xdx

x^2 +a^2 =

1

2 ln

(

x^2 +a^2

)

1.4

∫ dx

x^2 −a^2

=^1

2 a

ln

∣∣

∣∣x−a

x+a

∣∣

∣∣

1.5

∫ xdx

x^2 −a^2 =

1

2 ln

∣∣

x^2 −a^2

∣∣

2.Radicals

2.1

∫ dx

√

x^2 +a^2

=ln

(

x+

√

x^2 +a^2

)

or sinh−^1

(

x

a

)

2.2

∫ xdx

√

x^2 +a^2

=

√

x^2 +a^2

2.3

∫ dx

√

x^2 −a^2

=ln

(

x+

√

x^2 −a^2

)

(x>a)

2.4

∫ xdx

√

x^2 −a^2

=

√

x^2 −a^2

2.5

∫ dx

√

a^2 −x^2

=sin−^1

(

x

a

)

(|x|<a)

2.6

∫ xdx

√

a^2 −x^2

=−

√

a^2 −x^2 (|x|<a)

3.Exponentials and hyperbolic functions

3.1

∫

ekxdx=

ekx

k