The problem asks to evaluate the definite integral: $J = \int_0^{\frac{\pi}{2}} \cos(x) \sin^4(x) \, dx$

1. Problem Description

The problem asks to evaluate the definite integral:

$J = \int_0^{\frac{\pi}{2}} \cos(x) \sin^4(x) \, dx$

2. Solution Steps

We can solve this integral using a simple substitution.

Let $u = \sin(x)$. Then $\frac{du}{dx} = \cos(x)$, so $du = \cos(x) \, dx$.

Now we need to change the limits of integration.

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

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

So the integral becomes:

$J = \int_0^1 u^4 \, du$

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 $\frac{1}{5}$.