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The integral of the function $x^4 \cos(x)$ can be expressed as:

$$\int x^4 \cos(x) \, dx = x^4 \sin(x) + 4x^3 \cos(x) + 12x^2 \sin(x) - 24x \cos(x) - 24 \sin(x) + C,$$

where $C$ is the constant of integration.

To find this integral, we will apply the method of integration by parts. We start by letting:

• $u = x^4$
• $dv = \cos(x) \, dx$

Next, we compute the derivatives and integrals:

• $\frac{du}{dx} = 4x^3$
• $v = \sin(x)$

Using the integration by parts formula, which states that:

$$\int u \, dv = uv - \int v \, du,$$

we can express the integral as follows:

$$\int x^4 \cos(x) \, dx = x^4 \sin(x) - \int 4x^3 \sin(x) \, dx.$$

Now, we apply integration by parts again on the new integral $\int 4x^3 \sin(x) \, dx$. We set:

• $u = 4x^3$
• $dv = \sin(x) \, dx$

Calculating the necessary derivatives and integrals gives us:

• $\frac{du}{dx} = 12x^2$
• $v = -\cos(x)$

Substituting these back into the integration by parts formula yields:

$$\int 4x^3 \sin(x) \, dx = 4x^3 (-\cos(x)) - \int -\cos(x) \, (12x^2) \, dx = -4x^3 \cos(x) + \int 12x^2 \cos(x) \, dx.$$

Now substituting this back into our previous expression, we have:

$$\int x^4 \cos(x) \, dx = x^4 \sin(x) - 4x^3 \cos(x) + \int 12x^2 \cos(x) \, dx.$$

We repeat the integration by parts for $\int 12x^2 \cos(x) \, dx$. Let:

• $u = 12x^2$
• $dv = \cos(x) \, dx$

Calculating gives us:

• $\frac{du}{dx} = 24x$
• $v = \sin(x)$

This leads to:

$$\int 12x^2 \cos(x) \, dx = 12x^2 \sin(x) - \int 24x \sin(x) \, dx.$$

Substituting this result back, we have:

$$\int x^4 \cos(x) \, dx = x^4 \sin(x) - 4x^3 \cos(x) + 12x^2 \sin(x) - \int 24x \sin(x) \, dx.$$

Now, we apply integration by parts one last time for $\int 24x \sin(x) \, dx$. We set:

• $u = 24x$
• $dv = \sin(x) \, dx$

Calculating gives us:

• $\frac{du}{dx} = 24$
• $v = -\cos(x)$

Thus, we obtain:

$$\int 24x \sin(x) \, dx = 24x (-\cos(x)) - \int -\cos(x) (24) \, dx = -24x \cos(x) + 24 \sin(x).$$

Putting this back into our integral, we get:

$$\int x^4 \cos(x) \, dx = x^4 \sin(x) - 4x^3 \cos(x) + 12x^2 \sin(x) - (-24x \cos(x) + 24 \sin(x)).$$

After simplifying, we arrive at the final result:

$$\int x^4 \cos(x) \, dx = x^4 \sin(x) + 4x^3 \cos(x) + 12x^2 \sin(x) - 24x \cos(x) - 24 \sin(x) + C.$$

Here, $C$ is the constant of integration.