It is given that $$t>0$$ $$f(0)=1$$ $$f'(0)=0$$ and $$tf''(t)+f'(t)+tf(t)=0$$
From this is can be deduced that $\int_0^\infty e^{-at}f(t)dt=\frac{1}{\sqrt{a^2+1}}$
We then have to find the integral in question.
I really am not sure where to start? Is there something simple I am not seeing?