Wave eq: $y_{tt} = 4y_{xx}$
Domain: -infinity < $x$ < infinity
Initial conditions: $y(x,0)=cosx$ and $y_t(x,0)=\frac{ x}{x^2+1}$
As it's an infinite domain I think the d'Alembert solution must be used:
$$u(x,t)=\dfrac{f(x-ct)+f(x+ct)}{2}+\dfrac{1}{2c}\int_{x-ct}^{x+ct}g(s)ds$$
However I don't actually know how to apply this solution. Any help please?
You need to specify what's $f$ and what's $g$. I assume $f(x)=y(x,0)$ and $g(x)=y_t(x,0)$. Also, you need to compare your equation with the one stated in your textbook. I assume that $c^2=4$, so $c=2$. Then plugging into d'Alembert's formula results in $$u(x,t)=\frac{\cos(x-2t)+\cos(x+2t)}{2}+\frac1{4}\int_{x-2t}^{x+2t}\frac{x}{x^2+1}dx.$$ Finding an anti-derivative of $g$ by the method of substitution (left as homework), we end up with $$u(x,t)=\frac{\cos(x-2t)+\cos(x+2t)}{2}+\frac1{8}\ln (x^2+1)\Big|_{x-2t}^{x+2t}.$$
Regarding infinite domain: If it wasn't infinite, you would need to fix boundary conditions, and d'Alembert's formula wouldn't apply in a straight-forward way. See http://www.math.utk.edu/~freire/teaching/m435s14/PDE-Notes2013.pdf. Section 5 treats boundary conditions.