Consider these two definition of degree of a map $f : M \to N$ between closed connected and oriented differentiable manifolds of the same dimension $n$:
1)Assume that $f$ is continuous. Since $H_n(M) \cong \mathbb{Z}$ and $H_n(N) \cong \mathbb{Z}$, one must have $f_*([M]) = d[N]$, where $[M]$ and $[N]$ are generators for the respective homology groups. The integer $d$ is the degree of $f$. Denote it by $d(f)$.
2)Assume $f$ is smooth. Since $H_{dR}^n(M) \cong \mathbb{R}$ and $H_{dR}^n(N) \cong \mathbb{R}$ via integration (where these are the de Rham cohomology groups), there must exist a number (real, a priori) $\deg f$ such that $$\int_M f^* \alpha = \deg f \int_N \alpha,$$ for any smooth $n$-form $\alpha$ on $N$.
How does one show that these two definitions coincide when $f$ is smooth?