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Examples 10.1 to 10.20
Published in L. M. B. C. Campos, Classification and Examples of Differential Equations and their Applications, 2019
substitution of (10.208a) in (10.208b) specifies the remaining constant of integration (10.208c); substitution of (10.195a, b; 10.208a, c) in (10.207c) yields: ζ¯04(x)=qx2120EI{x3−10xL2+20L3},
Transport in Fluidic Nanochannels
Published in Victor M. Starov, Nanoscience, 2010
The transition from integration over the cross-sectional area to integration over the channel width in Equations 9.51 and 9.52 implies that the flow in the slit is considered to be two dimensional. The dimension that is parallel to the walls and normal to the flow direction is assumed to be infinite. The first integral of the PB equation 9.7 for a slit-shaped channel allows introducing the integration variable substitution d(zψ˜)κ2[cosh(zψ˜)−cosh(zψ˜m)=dx.
Systematic integration
Published in Alan Jeffrey, Mathematics, 2004
Integration by substitution depends for its success on transforming an integrand in one variable into one of simpler form in another variable. The choice of the substitution is dictated by experience, as there are no definite rules which can be formulated. As a general rule, a substitution is chosen which leads to the simplification of some difficult feature of the integrand, like a square or a square root. Sometimes more than one substitution is necessary in order to determine an antiderivative, and often integration by substitution needs to be combined with a different integration technique, such as partial fractions or integration by parts, each of which will be discussed later.
Why does trigonometric substitution work?
Published in International Journal of Mathematical Education in Science and Technology, 2018
Suppose that ∫f(x) dx is an integral for which integration by substitution and by parts do not apply. We will now present a theorem that sometimes can be used to evaluate such an anti-derivative. Since students, at this stage, are typically familiar with the derivatives of inverse functions, the theorem and its proof can easily be presented to students prior to (or after) addressing integration by trigonometric substitution. We now present a general result that confirms the correctness of integration by inverse substitution.
Don't throw the student out with the bathwater: online assessment strategies your class won't hate
Published in International Journal of Mathematical Education in Science and Technology, 2022
Stuart Johnson, John Maclean, Raymond F. Vozzo, Adrian Koerber, Melissa A. Humphries
Typically, an exam question that is too complicated for an online tool is also too complicated for a student. However, it is possible to test very specific aspects of a method on such a problem. For example, Figure A3 demonstrates an online-appropriate integration by substitution on an integral that is not possible to calculate in closed form. Thus an online calculator will give up and give a numerical answer, yet it is possible for a student to demonstrate basic knowledge of the method of substitution.