I am trying to become familiar with stochastic integration and stochastic differential notation. I tried to do the following little exercise. In my lecture notes the risky Asset is defined in the following way $$ \begin{align*} (1) &\qquad S(t)=S(0)*e^{ \int_{0}^{t} (r(s) \lambda (s)-\frac{1}{2} \sum_{n=1}^N \sigma_{n}^2(s) \,)ds + \sum_{n=1}^N\int_{0}^t\sigma_{n}(s)dW_n(s)} \\ (2) &\qquad dS(t)=S(t)[r(t)+\lambda(t)]dt+S(t)\sum_{n=1}^N\sigma_n(t)dW(t) \end{align*}$$
When I try to show (2) by using (1) and Itô's Lemma I get following Calculation $f(x):=S(0)e^{x}$,
$$R(t):=\int_{0}^{t} (r(s) \lambda (s)-\frac{1}{2} \sum_{n=1}^N \sigma_{n}^2(s) \,)ds + \sum_{n=1}^N\int_{0}^t\sigma_{n}(s)dW_n(s)\\$$ $$dS(t)=df(R(t))=S(t)dR(t)+\frac{1}{2}S(t)dR(t)dR(t)\\=S(t)[r(t) \lambda (t)-\frac{1}{2} \sum_{n=1}^N \sigma_{n}^2(t) \,]dt+\sum_{n=1}^NS(t)\sigma_{n}(t)dW_{n}(t)+\frac{1}{2}S(t)\sum_{n=1}^N\sigma_{n}^2(t)dt\\=S(t)r(t)\lambda(t)dt+\sum_{n=1}^NS(t)\sigma_{n}(t)dW_{n}(t)\\$$
Where is my mistake ? In equation$(2)$ the $S(t)$ is multiplicated with the sum of some Integrals with respect to the Brownian motion. If I do the Calculations, the $S(t)$ is in the sum and is integrated w.r.t the Brownian motion.