Question

evaluate d/dx ∫upper bound:5 lower bound:x g(t) dt

Answer 1

possible answers being :

-g(5)

-g(x)

g(5)

g(x)

Answer 2

This is the fundamental theorem of calculus. You can transform the integral by getting in the form \(\large \int_{a}^x\) by switching in integral limits and adding a negative sign. In fact, you can do this easily by first doing the integral, and the differentiating:
Let \(G(t)\) be the antiderivative of \(g(t)\)
\[\large \frac{d}{dx}\int_{x}^5g(t)dt\\=\frac{d}{dx}\int_{5}^x-g(t)dt\\=\frac{d}{dx}\left.-G(t)\right|_5^x \\=\frac{d}{dx}\left(-[ G(x)-G(5)]\right)\\
=\frac{d}{dx}(-G(x)+G(5))\\
=-g(x)\] since G(5) is just a number, the derivative of a number is 0

Answer 3

thank you !!!!
:D