Derivative of a constant integral

Question

I have the integral: $$\int_0^2 \frac{d}{dx}\left(\int_1^5\left(e^{\sqrt{x}}+x^3\right)dx\right)dx$$

This is zero right? Because the derivative of a constant is zero doesn't matter if its an integral, right?

Answer 1

That is a poorly written question, in that the variables of integration are the same.

Here is my transcription of the problem:

$$\int_0^2 \frac{d}{dx} \left( \int_1^5 \left( e^{\sqrt{x}}+x^3\right) dx \right) dx $$

Since the inner integral does not have any unbound variables (everything is either "$x$" or a constant), it is a constant. Therefore, its derivative is zero.

The outer integral is thus the integral of zero and, as such, is also zero.