EVALUATION OF THE NON-ELEMENTARY INTEGRAL ∫eλx^α, α≥2, AND OTHER RELATED INTEGRALS
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Bibliographic record
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 \(_{1}F_1\) by expanding the integrand as a Taylor series. 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 \(_{1}F_1\) 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 \(_1F_1\). Some of the applications of the non-elementary integral \(\int e^{\lambda x^\alpha} dx, \alpha\ge 2\) such as the Gaussian distribution and the Maxwell-Bortsman distribution are given.
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Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.005 | 0.005 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.001 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.003 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from it