1. **Problem statement:** Evaluate the integral $$\int_0^{\frac{\pi}{2}} \sin^4 \theta \cos^3 \theta \, d\theta$$ and express it in terms of the Beta function, then compute its value. 2. **Recall the Beta function definition:** The Beta function is defined as $$B(x,y) = 2 \int_0^{\frac{\pi}{2}} (\sin \theta)^{2x-1} (\cos \theta)^{2y-1} d\theta$$ for $x > 0$, $y > 0$. 3. **Match the integral to the Beta function form:** Given the integral $$\int_0^{\frac{\pi}{2}} \sin^4 \theta \cos^3 \theta \, d\theta$$ we rewrite powers as $$\sin^4 \theta = (\sin \theta)^4 = (\sin \theta)^{2 \cdot 2} = (\sin \theta)^{2 \cdot 2}$$ $$\cos^3 \theta = (\cos \theta)^3 = (\cos \theta)^{2 \cdot 1 + 1}$$ To fit the Beta function form, powers must be $2x-1$ and $2y-1$: - For sine: $2x - 1 = 4 \implies 2x = 5 \implies x = \frac{5}{2}$ - For cosine: $2y - 1 = 3 \implies 2y = 4 \implies y = 2$ 4. **Express the integral using Beta function:** $$\int_0^{\frac{\pi}{2}} \sin^4 \theta \cos^3 \theta \, d\theta = \frac{1}{2} B\left(\frac{5}{2}, 2\right)$$ 5. **Use the relation between Beta and Gamma functions:** $$B(x,y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x+y)}$$ 6. **Calculate Gamma values:** - $\Gamma\left(\frac{5}{2}\right) = \frac{3}{2} \cdot \frac{1}{2} \cdot \sqrt{\pi} = \frac{3 \sqrt{\pi}}{4}$ - $\Gamma(2) = 1! = 1$ - $\Gamma\left(\frac{5}{2} + 2\right) = \Gamma\left(\frac{9}{2}\right) = \frac{7}{2} \cdot \frac{5}{2} \cdot \frac{3}{2} \cdot \frac{1}{2} \cdot \sqrt{\pi} = \frac{105 \sqrt{\pi}}{16}$ 7. **Substitute into Beta function:** $$B\left(\frac{5}{2}, 2\right) = \frac{\frac{3 \sqrt{\pi}}{4} \times 1}{\frac{105 \sqrt{\pi}}{16}} = \frac{3/4}{105/16} = \frac{3}{4} \times \frac{16}{105} = \frac{48}{420} = \frac{4}{35}$$ 8. **Final integral value:** $$\int_0^{\frac{\pi}{2}} \sin^4 \theta \cos^3 \theta \, d\theta = \frac{1}{2} \times \frac{4}{35} = \frac{2}{35}$$ **Answer:** $$\boxed{\frac{2}{35}}$$