Answer
Part a) $e^{\pi/4}-1$
Part b) $0$
Part c) $e^{\arctan x}$.
Work Step by Step
Part a)
Using the Fundamental Theorem of Calculus in Part 2, $\int_0^1\frac{d}{dx}(e^{\arctan x}) dx=e^{\arctan 1}-e^{\arctan 0}=e^{\frac{\pi}{4}}-e^0=e^{\pi/4}-1$
Part b)
Since $\int_0^1e^{\arctan x}dx$ is a constant, using the derivative rule for a constant, we get $\frac{d}{dx}\int_0^1e^{\arctan x}dx=0$.
Part c)
Using the Fundamental Theorem of Calculus in Part 1, we get $\frac{d}{dx}\int_0^xe^{\arctan t}dt=e^{\arctan x}$.
