Substitution of a definite integral

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SUMMARY

The discussion focuses on the substitution method for evaluating a definite integral, specifically addressing the disappearance of the term √v during the integration process. The key point is that dv is expressed as (2/3)v^(-1/2)du, which clarifies the transformation in the integrand. The value 20/3 is derived from the multiplication of 10 by (2/3) during the substitution step, rather than from the integration itself.

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bobsmith76
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Homework Statement



Screenshot2012-02-04at91000PM.png


Screenshot2012-02-04at90922PM.png


The Attempt at a Solution



I don't see why the √v disappears in this step

Screenshot2012-02-04at91249PM.png


I understand how they got the 20/3, because they integrated, but if they're integrating then I would think v^-1/2 would become (v^1/2)/(1/2)
 
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bobsmith76 said:

Homework Statement



Screenshot2012-02-04at91000PM.png


Screenshot2012-02-04at90922PM.png


The Attempt at a Solution



I don't see why the √v disappears in this step

Screenshot2012-02-04at91249PM.png


I understand how they got the 20/3, because they integrated, but if they're integrating then I would think v^-1/2 would become (v^-1/2)/(1/2)

The √v disappears because [itex]dv = \frac{2}{3}v^{-\frac{1}{2}}du[/itex] (rearranging the equation they gave for du). Substitute that for dv into the integrand and you'll see why. The 20/3 comes from 10*(2/3) during this substitution, NOT from integration (which hasn't occurred yet).
 

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