Question:

Step 1: Evaluate the numerator integral. We need to compute $\displaystyle\int_{0}^{\pi/2}(5\cos^2 x + 3\sin^2 x)\,dx$. We can split this into two separate integrals: $$\int_{0}^{\pi/2}(5\cos^2 x + 3\sin^2 x)\,dx = 5\int_{0}^{\pi/2}\cos^2 x\,dx + 3\int_{0}^{\pi/2}\sin^2 x\,dx$$ Using the standard formulas $\displaystyle\int_{0}^{\pi/2}\cos^2 x\,dx = \frac{\pi}{4}$ and $\displaystyle\int_{0}^{\pi/2}\sin^2 x\,dx = \frac{\pi}{4}$: $$5 \cdot \frac{\pi}{4} + 3 \cdot \frac{\pi}{4} = \frac{5\pi + 3\pi}{4} = \frac{8\pi}{4} = 2\pi$$ Step 2: Evaluate the denominator integral using substitution. We need to compute $\displaystyle\int_{0}^{\pi/2}\sin\theta\cos\theta\sqrt{25\sin^2\theta + 9\cos^2\theta}\,d\theta$. Let $u = \sin^2\theta$, so $du = 2\sin\theta\cos\theta\,d\theta$, which means $\sin\theta\cos\theta\,d\theta = \frac{1}{2}du$. When $\theta = 0$: $u = 0$; when $\theta = \frac{\pi}{2}$: $u = 1$. Also, $\cos^2\theta = 1 - \sin^2\theta = 1 - u$. The integral becomes: $$\int_{0}^{\pi/2}\sin\theta\cos\theta\sqrt{25\sin^2\theta + 9\cos^2\theta}\,d\theta = \frac{1}{2}\int_{0}^{1}\sqrt{25u + 9(1-u)}\,du$$ Step 3: Simplify the expression under the square root. $$25u + 9(1-u) = 25u + 9 - 9u = 16u + 9$$ So the integral becomes: $$\frac{1}{2}\int_{0}^{1}\sqrt{16u + 9}\,du$$ Step 4: Evaluate the integral using the power rule. Let $w = 16u + 9$, so $dw = 16\,du$, which means $du = \frac{1}{16}dw$. When $u = 0$: $w = 9$; when $u = 1$: $w = 25$. $$\frac{1}{2}\int_{0}^{1}\sqrt{16u + 9}\,du = \frac{1}{2} \cdot \frac{1}{16}\int_{9}^{25}w^{1/2}\,dw$$ $$= \frac{1}{32}\left[\frac{2}{3}w^{3/2}\right]_{9}^{25} = \frac{1}{48}\left[w^{3/2}\right]_{9}^{25}$$ $$= \frac{1}{48}\left[25^{3/2} - 9^{3/2}\right] = \frac{1}{48}\left[125 - 27\right] = \frac{98}{48} = \frac{49}{24}$$ Step 5: Calculate the ratio of numerator to denominator. $$\frac{\text{Numerator}}{\text{Denominator}} = \frac{2\pi}{\frac{49}{24}} = 2\pi \cdot \frac{24}{49} = \frac{48\pi}{49}$$ **Final Answer:** The value of the given expression is $\boxed{\dfrac{48\pi}{49}}$, which corresponds to **Option 2**.