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Derivative (Integral) = Integral (Derivative)? (Posted on 2012-10-24) Difficulty: 1 of 5
Is d/dx (∫ f(x) dx) equal to ∫ d/dx (f(x)) dx ?

Give reasons for your answer.

No Solution Yet Submitted by K Sengupta    
Rating: 2.5000 (2 votes)

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Solution No joke, but a new reason Comment 3 of 3 |
Consider the Dirichlet function: f(x) = 0 if x is rational, 1 otherwise

This function has no derivatives, so integral(derivative) does not exist.

However, I believe its Lebesque integral is x (plus a constant).  The derivative of its integral = 1, which is not the same as f(x).

So, f(x), the derivative of the integral and the integral of the derivative can all be different.

Of course, this problem is only difficulty = 1, so I expect that this is not the expected answer.

  Posted by Steve Herman on 2012-10-25 02:30:39
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