I wonder how to solve the Klein-Gordon equation $$\left\{\begin{split}&\partial_t^2u-\Delta u+u=f\\&u(0,x)=u_1(x),\quad \partial_t u(0,x)=u_2(x)\end{split}\right.$$ where $u(t,x)$ defined on $\mathbb{R}^+\times\mathbb{R}^d$.
By applying the partial Fourier transform respect to the $x$ variable, I obtain the following representation of u as $$\begin{split}u(t,x)=&\mathcal{F}^{-1}(\cos\sqrt{1+|\xi|^2}t)*u_1(x)+\mathcal{F}^{-1}(\frac{\sin\sqrt{1+|\xi|^2}t}{\sqrt{1+|\xi|^2}})*u_2(x)\\&+\int_0^t\mathcal{F}^{-1}(\frac{\sin\sqrt{1+|\xi|^2}(t-s)}{\sqrt{1+|\xi|^2}})*f(s,x)\,ds\end{split}$$$u$: However$$\begin{split}u(t,x)=&\mathcal{F}^{-1}\left(\cos\sqrt{1+|\xi|^2}t\right)*u_1(x)+\mathcal{F}^{-1}\left(\frac{\sin\sqrt{1+|\xi|^2}t}{\sqrt{1+|\xi|^2}}\right)*u_2(x)\\&+\int_0^t\mathcal{F}^{-1}\left(\frac{\sin\sqrt{1+|\xi|^2}(t-s)}{\sqrt{1+|\xi|^2}}\right)*f(s,x)\,ds\end{split}$$
However, I don't know how to solve the inverse Fourier transforms in the formula.: do there exist closed form expressions for such integrals?
