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  • Solving Integrals for e^-ax^2: (i), (ii) (iii) - Physics Forums
    The discussion revolves around evaluating integrals of the form \ (\int_ {0}^ {\infty} e^ {-ax^2} x^n dx\) for \ (n = 2, 3, 4\), given the known integral \ (\int_ {0}^ {\infty} e^ {-ax^2} dx = \frac {\sqrt {\pi}} {2\sqrt {a}}\) Participants explore methods such as differentiation with respect to the parameter \ (a\) and integration by parts Participants discuss using differentiation with
  • Understanding Integration of 1 (x^2 + a^2) and the Role of the Tan . . .
    2) use the substitution , which gives , the indefinite integral you already solved (up to a constant 1 a) Just to be clear, all my integrals here are indefinite integrals
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    The discussion revolves around the relationship between integrals and the area of geometric shapes, specifically focusing on the area of half a circle and an ellipse Participants explore the implications of certain integrals and their geometric interpretations Exploratory, Conceptual clarification, Mathematical reasoning, Problem interpretation Participants discuss the integral \ (\int
  • How do you derive the pV Work formula, W= integralpdV?
    This discussion revolves around deriving the pV Work formula, specifically W = ∫pdV, within the context of thermodynamics Participants are exploring the relationship between force, pressure, and work done during a state change Conceptual clarification, Mathematical reasoning, Problem interpretation Participants discuss substituting force in the work equation and the implications of
  • How can I solve integrals of the form x^n e^ (-x^2) by hand?
    Making substitution for x^2 still leaves a factor x in the denominator since u = x^2 implies du = 2 x dx The result then would give a square root of u in the integral, which I cannot solve by integration by parts Is there any way to find an explicit formula for this integral?





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