Dr. D. K. Yadav
Chapter-II: Six Conjectures on Indefinite Nonintegrable Functions
Six Conjectures on Indefinite Nonintegrable Functions or Nonelementary Functions
Dharmendra Kumar Yadav Assistant Professor, Department of Mathematics Shivaji College, University of Delhi, Raja Garden, Delhi-27
[email protected]
+91-9891643856
Note: These six conjectures are part of my Ph. D. work. I have discussed these with proof. At last I have provided their possible solutions after introducing dominating sequential functions. The contents in this chapter have been published in 7 parts as given in next page:
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Dr. D. K. Yadav
Chapter-II: Six Conjectures on Indefinite Nonintegrable Functions
Conjecture-1: An indefinite integral of the form
e f ( x) dx , where f(x) is a polynomial f ' ( x)
function of degree ≥2, or a trigonometric (not inverse trigonometric) function, or a hyperbolic (not inverse hyperbolic) function is always nonintegrable.
Conjecture-2: An indefinite integral of the form
e f (x) dx , where f(x) and g(x) are polynomial g ( x)
functions in x of degree greater than or equal to 1, is always nonintegrable.
Conjecture-3: An indefinite integral of the form
f ( x) dx ; where f(x) is a trigonometric (not g ( x)
inverse trigonometric) function, or a hyperbolic (not inverse hyperbolic) function, and g(x) is a polynomial of degree greater than or equal to 1, is always nonintegrable.
Conjecture-4: An indefinite integral of the form e f ( x ) dx , where f(x) is a trigonometric (not inverse trigonometric) function, or a hyperbolic (not inverse hyperbolic) function, or a polynomial of degree greater than or equal to 2, is always nonintegrable.
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Dr. D. K. Yadav
Chapter-II: Six Conjectures on Indefinite Nonintegrable Functions
Conjecture-5: An indefinite integral of the form g [ f ( x)]dx , where f(x) is a polynomial of degree greater than or equal to 2 and g(x) is a trigonometric (not inverse trigonometric) or a hyperbolic (not inverse hyperbolic) function is always nonintegrable.
Conjecture-6: An indefinite integral of the form
f ( x).g ( x) dx , where f(x), h(x) are h ( x)
polynomials in x (degree of h(x) is greater than the degree of f(x)) and g(x) is a trigonometric (not inverse trigonometric) or a hyperbolic (not inverse hyperbolic) function is always nonintegrable.
References: D. K. Yadav, General Study on Non-integrable Functions in Indefinite Integral Case, International Research Journal, Acta Ciencia Indica, Vol.33M, No.4, 1667-1670, 2007, Pragati Prakashan, Meerut, U.P., INDIA, ISSN: 0970-0455. Mathematical Reviews, USA, MR2442861 D. K. Yadav and D.K.Sen, Revised Paper on Indefinite Nonintegrable Functions, International Research Journal, Acta Ciencia Indica, Vol.34 M, No.3, 1383-1384, 2008, Pragati Prakashan, Meerut, U.P., INDIA, ISSN: 0970-0455 D. K. Yadav & D. K. Sen, Proof of First Standard Form of Non-elementary Functions, I. J. of Advanced Research in Science & Engineering, 2(2), 1-14, February 2013,Online: www.ijarse.com, A. R. Research Publication, New Delhi, ISSN: 2319-8354(E) D. K. Yadav & D. K. Sen, Proof of Second Standard Form of Nonelementary Functions, I. J. of Advanced Research in Comp. Science & Soft. Engineering, 3(2), 103-105, 2013, Online: www.ijarcsse.com, Advance Research International Publication House, Jaunpur, U.P., ISSN: 2277-128X
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Dr. D. K. Yadav
Chapter-II: Six Conjectures on Indefinite Nonintegrable Functions
D. K. Yadav & D. K. Sen, Proof of Third & Sixth Standard Forms of Nonelementary Functions, Int. Journal of Advanced Research in Computer Science & Software Engineering, 3(4), 247-257, 2013, Online: www.ijarcsse.com, Advance Research Int. Publication House, Jaunpur, U.P., ISSN: 2277-128X D. K. Yadav & D. K. Sen, Proof of Fourth Standard Form of Nonelementary Functions, I. J. of Advanced Research in Comp. Science & Soft. Engineering, 3(4), 258-264, 2013, Online: www.ijarcsse.com, Advance Research International Publication House, Jaunpur, U.P., ISSN: 2277-128X D. K. Yadav & D. K. Sen, Proof of Fifth Standard Form of Nonelementary Functions, I. J. of Advanced Research in Comp. Science & Soft. Engineering, 3(4), 269-274, 2013, Online: www.ijarcsse.com, Advance Research International Publication House, Jaunpur, U.P., ISSN: 2277-128X
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