Answer
$$f(\theta)
=76-\frac{5}{2} \sin^2\theta -9\sin \theta.
$$
Work Step by Step
We have
$$f(\theta)=\int_{\sin \theta}^{4}5t+9 \ dt =\frac{5}{2}t^2 +9t|_{\sin \theta}^{4}\\
=40-\frac{5}{2} \sin^2\theta+36-9\sin \theta\\
=76-\frac{5}{2} \sin^2\theta -9\sin \theta.
$$
