We need to evaluate the definite integral $J = \int_{0}^{\frac{\pi}{2}} \cos x \sin^4 x \, dx$.

1. Problem Description

We need to evaluate the definite integral $J = \int_{0}^{\frac{\pi}{2}} \cos x \sin^4 x \, dx$.

2. Solution Steps

Let $u = \sin x$. Then, $du = \cos x \, dx$.

When $x = 0$, $u = \sin 0 = 0$.

When $x = \frac{\pi}{2}$, $u = \sin \frac{\pi}{2} = 1$.

So, we can rewrite the integral in terms of $u$ as:

$J = \int_{0}^{1} u^4 \, du$

Now, we can evaluate the integral:

$J = \left[ \frac{u^5}{5} \right]_{0}^{1}$

$J = \frac{1^5}{5} - \frac{0^5}{5}$

$J = \frac{1}{5} - 0$

$J = \frac{1}{5}$

3. Final Answer

The final answer is $\frac{1}{5}$.