The exercise asks to show that the function $x\mapsto e^{-|x|^\alpha}$ from $\mathbb{R}^d$ to $\mathbb{R}$ is is $\lambda^d$ integrable for every $\alpha>0$, where $\lambda^d$ denotes Lebesgue measure on $\mathbb{R}^d$. As a hint we are referred to a previous exercise where we have shown that the same function on $\mathbb{R}$ is $\lambda^1$ integrable.
This question uses polar coordinates, but in my book we haven't used this technique yet. Rather I think we must use Tonelli's theorem, but then how can I show integrability of each of the $d$ integrals over $\mathbb{R}$?
This can be done with Fubini-Tonelli's theorem. Let $f : \mathbb{R}^d \to \mathbb{R}$ be a function such that
$$0 \leq f(x_1, \ldots, x_d) \leq \prod_{i=1}^d g_i (x_i)$$
for all $(x_i)_{1 \leq i \leq d}$ and some non-negative functions $g_i$. Then Fubini-Tonelli's theorem let us split the integral of $f$:
$$0 \leq \int_{\mathbb{R}^d} f \ d \lambda^d \leq \prod_{i=1}^d \int_{\mathbb{R}} g_i \ d \lambda^1.$$
Now, it is enough to find an integrable function $g$ such that $e^{-|x|^\alpha} \leq \prod_{i=1}^d g(x_i)$ everywhere.
The simplest thing to try is to take $g(x_i) = e^{-c |x_i|^\alpha}$ for some constant $c > 0$ (which may depend on $d$). By monotonicity, the inequality holds if and only if
$$|x|^\alpha \geq c \sum_{i=1}^d |x_i|^\alpha$$
for all $x \in \mathbb{R}^d$. This can be done (for instance with $c = 1/d$), but at this point, I'll let you try.