Evaluate the integral given below: $$ \int_0^4 \frac{\sin \sqrt{t}}{\sqrt{t}} dt$$
I tried evaluating this integral by using substitution theorem for integrals( by defining $f(x)= 2\sin(x)$ and $\Phi(t)= t^{1/2}$ but turns out when defining $\Phi$, the derivative doesn't exist at zero and so I can't really apply the substitution here.
And so I thought if I could apply Fundamental theorem for calculus on the function $f(x)$, that $f$ is integrable for $(0,2]$ where $0=\Phi(0)$ and $2=\Phi(4)$, but then I got lost. I remember reading somewhere that if the function is bounded by $M$ and integrable on restricted intervals $[c,b]$ where $c$ is an interior point in $(a,b)$ then it becomes integrable on the whole interval $[a,b]$, that's what I thought will help me to apply Fundamental theorem (I think squeeze theorem might come in handy as well) but I still have no idea how will it help me to evaluate the value of the integral? I seem to be missing a lot here. Can I get a helpful hint or a solution to this problem?
P.S: I know the solution where I could simply use precalculus technique of substitution to solve but I need to prove that the integral exists using fundamental theorem for calculus given in real analysis.