Given that $\int_0^\infty e^{-at}f(t)dt=\frac{1}{\sqrt{a^2+1}}$, what is $\int_0^{\infty}f(t)dt?$

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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?