Differentiating using $\frac{d}{dx}$ of an integral of a function as function of $x$ with bounds including functions as function of $t$

Question

We know how how differentiate an integral using the Leibniz theorem, but how can we differentiate such integrals (shown in the image)?

Answer 1

If we were to evaluate it our completely we would get $\frac{t^2}{2}$ from a to $x^2.$ plugging in our bounds we get $\frac{x^4}{2}-\frac{a^2}{2}$ as we can see here there are no t's anywhere so taking the derivative with respect to t yields 0. Now let's say I had a function I can't integrate such as $\int_{0}^{x^2}e^{-t^2} \, \mathrm{d}t$ if we wanted to find $\frac{d}{dt}$ then it would be zero because even though we can't evaluate it directly, we know that the final answer will only involve x's and constants, making the derivative with respect to t 0

Answer 2

For the first one zero, once you evaluate you will get a function of just x whose derivative with respect to t is just 0. The second one is the exact same as the original example. You can either a) evaluate it out or b) say t=x dt=dx and then evaluate using second fundamental theorem