12-267/Tuesday September 11 Notes

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Solving the complicated integral in the Brachistochrone integral

Complete the square in the integrand:

Substitute and :

Assuming all variables are positive, substitute and . Then and :

For the integrand substitute and . Then transform the integrand using the substitutions , and :

Simplify the integrand to get :

For the integrand use partial fractions:

For the integrand , substitute and :

The integral of is :

Substitute back for :

Substitute back for :

Substitute back for :

Substitute back for :

Factor the answer a different way:

Which is equivalent for restricted y and d values to:

Syjytg 23:00, 11 September 2012 (EDT)