Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be given by $$ g(t)=\int_\left[0,1\right]e^\sqrt{x^2+t^2}d\lambda(x), t \in \mathbb{R} $$ a) Justify that the integral is defined for each $t \in \mathbb{R}$ and show next that g is continuous. b) Show that g is differentiable and compute $g'(t)$ for all $t \in \mathbb{R}$. Find in particular $g'(0)$
My approach for (a) was to use the continuity lemma of lebesgue integrals however I am obviously unable to find a function $w(x) \in \mathcal{L^1_+}:\left|u(t,x)\right|\leq w(x)\forall (t,x) \in \mathbb{R}\times [0,1]$
For (b) I run into the same problem since the partial derivative w.r.t. t isn't bounded either. I'm sorta suspecting that my teacher has made a mistake and forgot to include a minus before the squareroot, however I might be mistaken, thus I'm asking here.
Hint: You can't get a single $w$ that works for all $t \in \mathbb R$, but for any $K> 0$ you can get one that works for all $|t| \le K$.