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Integral, Continuity and Limits

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Q:
Suppose and are continuous and F(x) =
Let

(a) prove that f'(x) + g'(x) =0 for all x
(b) Prove that f(x) + g(x) = /4 for all x. Deduce that

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Solution Summary

The Leibniz derivative and the Poisson integral are applied. The solution is detailed and well presented.

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Suppose and are continuous and F(x) =
Let

(a) prove that f'(x) + g'(x) =0 for all x
Prove that f(x) + g(x) = /4 for all x. Deduce that

Solution:

(a) We will apply the formula of Leibniz for derivative of an integral with parameters:
( 1)
For f(x), we will have:
( 2)
where
...

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