Find the value of c ?

We can apply Laplace's method for integrals to deduce the asymptotic behaviour $\begin{aligned} \int_0^{\frac{\pi}{2}} x^r \sin x\,dx & \sim \; r^{-1}\big(\tfrac12\pi\big)^{r+1} \\ \int_0^{\frac{\pi}{2}} x^r \cos x\,dx & \sim \; r^{-2}\big(\tfrac12\pi\big)^{r+2} \end{aligned}$ as $r \to \infty$ , so that $\frac{\displaystyle r^c \int_0^{\frac{\pi}{2}} x^r \sin x\,dx}{\displaystyle \int_0^{\frac{\pi}{2}} x^r \cos x\,dx} \; \sim \; \frac{2r^{c+1}}{\pi} \hspace{2cm} r \to \infty$ and hence we deduce that $c = \boxed{-1}$ is the required value.