{
  "id": "1702.08438",
  "version": "v1",
  "published": "2017-02-25T01:01:08.000Z",
  "updated": "2017-02-25T01:01:08.000Z",
  "title": "Evaluation of the non-elementary integral $\\int e^{λx^α} dx, α\\ge2$, and related integrals",
  "authors": [
    "Victor Nijimbere"
  ],
  "comment": "15 pages, 1 figure",
  "categories": [
    "math.CA"
  ],
  "abstract": "A formula for the non-elementary integral $\\int e^{\\lambda x^\\alpha} dx$ where $\\alpha$ is real and greater or equal two, is obtained in terms of the confluent hypergeometric function $_1F_1$. This result is verified by directly evaluating the area under the Gaussian Bell curve, corresponding to $\\alpha = 2$, using the asymptotic expression for the confluent hypergeometric function and the Fundamental Theorem of Calculus (FTC). Two different but equivalent expressions, one in terms of the confluent hypergeometric function $_1F_1$ and another one in terms of the hypergeometric function _1F_2, are obtained for each of these integrals, $\\int \\cosh(\\lambda x^\\alpha)dx$, $\\int \\sinh\\lambda x^\\alpha)dx, $\\int \\cos(\\lambda x^\\alpha)dx and $\\int \\sin(\\lambda x^\\alpha)dx, $\\lambda\\in \\mathbb{C}$, \\alpha\\ge2$. And the hypergeometric function $_1F_2$ is expressed in terms of the confluent hypergeometric function $_1F_1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-25T01:01:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "26A36",
      "33C15",
      "30E15"
    ],
    "keywords": [
      "confluent hypergeometric function",
      "non-elementary integral",
      "related integrals",
      "evaluation",
      "gaussian bell curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}