or new section

( ) ( )

1

) ( li ln

ln ln ln

1 1

+

= ⁺ − ⁺

∫

^{xn x mdx xn x}_n^{m m xn}

For n=−1

( )

^{dx x}

[ ( )

^{x m}

]

x

x m

m

−

∫

^{ln ln} =^{ln ln ln}

• Math.com calculations; same approach as for above integral,

( )

ln ln

( )

ln li( ) ln x ⁿdx=x x ⁿ −n x

• For first integral, change of variable:

∫

( )

( ) ∫

^{( )}

∫

^{( )}

( )

∫

^{+ + +}

+ +

=

=

=

=

=

=

=

=

ds s m e ds s e

ds s x I

e x x e x e

ds dx x s

s n m

s m n

n

s n n x

s

ln ln

ln

, ,

, ln

1 1 1

1 1 ln

by relation logxⁿ =nlogx

• Then, int. by parts with,

( ) .

,

lns dv e ¹ ds m

u= = ^{n+ s}

NOTE: forn=0,m=n, see above integral,

( )

ln ln

( )

ln li( ) ln x ⁿdx=x x ⁿ −n x

∫

• For n=−1: change of variable &

int. by parts;

( )

( )

^x

( )

^{x m x}

x m dx x

x

sds m s s m s vds uv

s ds v ds dv

s ds s,du m m u

sds m x ds

s I x

x ds dx x s

m m

m

ln ln

ln ln ln

ln ln

ln ,

ln : parts by int.

. ) ln

ln(

, ln

−

=

−

=

−

=

−

=

=

=

=

=

=

=

=

=

∫

∫

∫

∫

∫

∫

—Sect. 2.71 & 2.72-2.73—

Section 2.724

( )

and substitute for solution of

( )

• Alternative statement of solution:

( ) ( )

based on converting

( ) produces a different ordering of the series terms.

NOTE: 2.724.2 follows if m = 1.

NOTE: m=2,3: substitute into expansion formula Formula 2.724.2

Change Formula 2.724.2 to read:

( )

NOTE: new version relates exponential- and logarithm-integrals, Ei

( )

x &li

( )

x, Formula 8.211.1.

—Sect. 2.71 & 2.72-2.73—

New Section

( ) ( )

Integration by parts with,

(

ln

)

, .

NOTE: a difficult (to reduce further) but general integral useful for deriving existing forms by setting

n m c b

a, , , , as desired.

New Section

( ) ( )

• Integration by parts:

( ) ( )

Integration by parts:

bx

• NOTE: a difficult (to reduce further) but general integral useful for deriving existing forms (such as 2.711, 2,722, 2.733, 2.726) by setting

n

—Sect. 2.71 & 2.72-2.73—

Sect. 2.727 or New Section

( ( ) )

• Integration by parts:

( )

^{x dv}

(

^{a bx}

)

^dx

New Section

( ) ( ) ( )

• Expansion formula:

™ Change of variable;

™ Now use indefinite integral previously derived (omitted in GR 7, but submitted as erratum),

( ) ( )

™ Writing summation in reverse order:

—Sect. 2.71 & 2.72-2.73—

NOTE: this formula may be derived readily by restating

( )

( )

and applying integral derived above in Sect. 2.32,

( )

with parameters

( ) Sect. 2.32, above, for multiple solutions to

( )

∫

^{ln a}_x^{+bx dx}(in terms of dilogarithms, defined in Math. Summary) all of which differ from equivalent GR solution!

• New reduction formula; same reduction approach as Formula 2.728.1

• Integration by parts:

( ) ( )

by Formula 8.240.1.

¾ Verified math.com

( )

—Sect. 2.71 & 2.72-2.73—

¾ verified math.com New Section See integral below,

( )

Substituting

—Sect. 2.71 & 2.72-2.73—

NOTE: easier approach: int. by parts on

( )

_n

derived by change of variable with

( )

¾ Mathematica provided solution in terms of hypergeometric function.

New Section

—Sect. 2.71 & 2.72-2.73—

Substituting

( ) ( )

(change of variable,

)

—Sect. 2.71 & 2.72-2.73—

• Integer-expansion formula:

From above,

Apply integral formula, above, Sect. 2.32,

( ) ( ) ( )

with parameters,

[

s= x=a−lnx,n=1,m=n,β =1,z=n−1

]

o Alternative expression of expansion formula:

—Sect. 2.71 & 2.72-2.73—

previously submitted as Form. 2.325.19, which produces a different ordering of the series terms.

• n=2: In above integral,

NOTE: Formula 4.212.4 results for this integral with limits from 0 to 1.

o Integration by parts:

o produces a poor reduction formula since powers of

x

ln are increasing. Setting n→−n in 2.724 produces desired results.

• For m = 1:

—Sect. 2.71 & 2.72-2.73— formula to solve Formula 2.721.3,

∫

_x^dx_lnx ^n==1li(1).

• For n=1:

( )

^ln =−

(

−¹

)( )

^{1ln −1}

∫

_x ^dx_x ^mdx _{m x} ^m by change of variable,

. ln x s=

• Integer-expansion formula:

Define from above,

∫

⁻( )⁻

with parameters

[

x=s=lnx,n=1,β =n−1,z=m−1

]

.

• Alternative form for expansion:

( )

⁼

based on first submission for expansion of

∫

^e−_x^{mx dx}

β n

,

Formula 2.325.19, which produces a different ordering of the series terms.

¾ EXAMPLES:

—Sect. 2.71 & 2.72-2.73—

• First two integrals: Similar to above integrals,

(

ln

)

^&

∫

^{ln .}

∫

_a−^dx_x ⁿ _a−^dx _x

• Integer-expansion formula Change of variable:

.

Apply above integral, Sect. 2.32,

( ) ( )

with parameters

[

^{x=s=a+lnx,n=1,m=n,a=1,z=n−1}

]

o Alternative form for expansion:

( )

( )( ) ( )( )

based on first submission for expansion of

∫

^e_x^{axm dx}

n

, Form. 2.325.8, which produces a different ordering of the series terms.

• n=2:Similar to above integral,

o NOTE: Formula 4.212.3 results for this integral with limits 0 to 1.

—Sect. 2.71 & 2.72-2.73—

New Section

( )

Change of variable:

( ) ( ) Sect. 2.32, above:

( )

( )

¾ Verified Mathematica

( )

∫

math.com calculation.

See derivation of definite integral, Sect. 4.212, below.

math.com calculation

See derivation of definite integral, Sect. 4.212, below.

∫

^e_x^{axm dx}

n

∫

^xmeaxndx

Express exponential integrals as equivalent integrals in logarithms.

NOTE: see Sect. 4.215, below, for similar definite integrals.

REFERENCES

Abramowitz, Milton and Irene Stegun. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Washington, DC: U.S.Government Printing Office, 1964 (Reprinted Dover, New York, 1972).

Artin, Emil. The Gamma Function. New York, NY: Holt, Rinehart and Winston, 1964.

Bers, Lipman. Calculus. 2 volumes. NY: Holt, 1969.

Boas, Mary L. Mathematical Methods in the Physical Sciences, 3 ed. NY: Wiley, 2006.

Boros, G. and Moll, V. Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals. Cambridge, England: Cambridge University Press, 2004.

Carr, G.S. Formulas and Theorems in Pure Mathematics, 2 ed., New York: Chelsea, 1970

Dwight, H.B. Table of Integrals and Other Mathematical Data, 4th ed. New York: Macmillan, 1961.

Gautschi, W. “The incomplete gamma functions since Tricomi.” In Tricomi's Ideas and

Contemporary Applied Mathematics, Atti dei Convegni Lincei, n. 147, Accademia Nazionale dei Lincei, Roma, 1998, pp. 203-237.

Gradshteyn, I.S. and I.M. Ryzhik (6^th Edition). Table of Integrals, Series, and Products. Alan Jeffrey and Daniel Zwillinger, Editors. NY: Academic Press, 2000.

_______ (7^th Edition). Table of Integrals, Series, and Products. Alan Jeffrey and Daniel Zwillinger, Editors. NY: Academic Press, 2007.

Moll, Victor H. The integrals in Gradshteyn and Rhyzik. Part 2: Elementary logarithmic integrals.

Scientia, Series A: Mathematical Sciences, Vol. 14 (2007), 7-15

_______. The integrals in Gradshteyn and Ryzhik. Part 3: Combinations of Logarithms and Exponentials. arXiv:0705.0175v1 [math.CA]. 1 May 2007

_______.The integrals in Gradshteyn and Ryzhik. Part 4: The Gamma function arXiv:0705.0179v1 [math.CA]. 1 May 2007.

_______. The integrals in Gradshteyn and Ryzhik. Part 13: Evaluation using The Error Function.

http://www.math.tulane.edu/~vhm/web_html/erfweb.pdf, 4 October 2006

_______. The integrals in Gradshteyn and Ryzhik. Part 14: The Exponential-Integral Function.

http://www.math.tulane.edu/~vhm/web_html/expintweb.pdf, 6 Semptember 2006.

_______. The integrals in Gradshteyn and Ryzhik. Part 15: The Incomplete Gamma Function http://www.math.tulane.edu/~vhm/web_html/erfweb.pdf, 4 September 2006

Moll, Victor H., Jason Rosenberg, Armin Straub, Pat Whitworth. The Integrals in Gradshteyn and Ryzhik. Part 8: Combinations of powers, exponentials and logarithms. arXiv:0707.2123v1

[math.CA]. 14 Jul 2007.

O’Brien, F.J. 100 Proposed Formulas Submitted for Publication in I.S. Gradshteyn and I.M. Ryzhik Table of Integrals, Series, and Products, 7^th Edition. Alan Jeffrey and Daniel Zwillinger, Editors.

NY: Academic Press, 2005. Newport, RI: Naval Undersea Warfare Center, Division, Newport.

November 1, 2005.

_______. Selected Transformations, Identities, and Special Values for the Gamma Function.

Submittted to Mathworld, http://functions.wolfram.com/GammaBetaErf/Gamma, June, 2006.

_______. 400 Proposed Formulas Submitted for Publication in I.S. Gradshteyn and I.M. Ryzhik Table of Integrals, Series, and Products, 8^th Edition. Alan Jeffrey and Daniel Zwillinger, Editors.

NY: Academic Press, 2007. Newport, RI: Naval Undersea Warfare Center, Division, Newport.

May 9, 2007.

_______. Summary of Four Generalized Exponential Models (GEM) For Continuous Probability Distributions, Jan. 18, 2008. arXiv:0801.2941v2 [math.GM]

Prudnikov, A. P., Yu. A. Brychkov, and O. I. Marichev. Integrals and Series. Vol. 1: Elementary Functions. Gordon & Breach, New York, 1986.

_______. Integrals and Series. Vol. 2: Special Functions. Gordon & Breach, New York, 1986.

_______. Integrals and Series. Vol. 3: More Special Functions. Gordon & Breach, New York, 1990.

Sokolnikoff, I. S. Advanced Calculus. New York: McGraw Hill Book Co., 1939.

Spiegel, M. R. Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 1968.

[Reprinted Spiegel, Murray R., John Liu, and Seymour Lipschutz. Mathematical Handbook of Formulas and Tables. New York: McGraw-Hill, 1999.]

Whittaker, E. T. and G. N. Watson. A Course of Modern Analysis. Cambridge University Press; 4th edition, 1934.

Wolfram Research, Inc. The Wolfram Functions Site. http://functions.wolfram.com/, 1998-2008.

_______. Wolfram Mathworld. http://mathworld.wolfram.com/, 2008.

Zwillinger, Daniel. Handbook of Integration. Boston: Jones & Bartlett Publishers, Inc., 1992.

INDEX OF SYMBOLS, FUNCTIONS,

In document 500 Integrals (Page 79-94)