Discussion Overview
The thread discusses various challenging integrals, specifically focusing on the integral of a function involving an exponential and a power of x, as well as the integral of the square root of sine cubed. Participants explore potential methods for solving these integrals, including substitutions and integration by parts, while expressing uncertainty about the solvability in elementary functions.
Discussion Character
- Exploratory
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant presents the integral \(\int \frac{3}{2}x^{-\frac{1}{2}}\cdot e^{\frac{3}{2}x^{-2}}\,dx\) and questions its solvability beyond using a Taylor series.
- Another suggests a substitution \(u = \sqrt{x}\), which simplifies the integral to one involving \(e^{u^{-4}}\).
- Concerns are raised about discrepancies in results from different computational tools, with one participant noting that Mathematica indicates the integral is unsolvable.
- Further discussion includes attempts to clarify the expression of the integral and the results obtained from Mathematica, with one participant mentioning a gamma function as part of the solution.
- A separate integral involving \(\int \sin^{\frac{3}{2}}(x)\;dx\) is introduced, with participants expressing uncertainty about its solvability and discussing potential substitutions.
- One participant claims to have derived an expression for the integral in terms of elementary functions using integration by parts and substitution.
- Another participant expresses admiration for the approach taken and seeks clarification on the substitution method used.
- There is a discussion about the creativity involved in solving integrals and the subjective nature of finding effective substitutions.
Areas of Agreement / Disagreement
Participants express differing views on the solvability of the integrals discussed, with some suggesting methods that may lead to solutions while others remain skeptical about the existence of elementary solutions. The discussion remains unresolved regarding the integrals' solvability.
Contextual Notes
Some participants mention limitations in their computational tools and the potential for errors in inputting expressions, which may affect the results obtained. The discussion also highlights the complexity of the integrals and the various approaches that may or may not yield satisfactory results.