Answer is: option4

\(\frac{x}{\sqrt{1 - x^2}}\)

Solution:

We are given the equation:

\[ \int \arccos(x) \, dx = x \arccos(x) + \int f(x) \, dx \]

Step 1: Use Integration by Parts for \( \int \arccos(x) \, dx \)

We will solve \( \int \arccos(x) \, dx \) using integration by parts. Let us choose:

\[ u = \arccos(x) \quad \Rightarrow \quad du = \frac{-1}{\sqrt{1 - x^2}} \, dx \]

\[ dv = dx \quad \Rightarrow \quad v = x \]

Step 2: Apply Integration by Parts

Using the integration by parts formula:

\[ \int u \, dv = u v - \int v \, du \]

Substitute the values of \( u \), \( du \), \( v \), and \( dv \):

\[ \int \arccos(x) \, dx = x \arccos(x) - \int x \cdot \frac{-1}{\sqrt{1 - x^2}} \, dx \]

Simplify the expression:

\[ \int \arccos(x) \, dx = x \arccos(x) + \int \frac{x}{\sqrt{1 - x^2}} \, dx \]

Step 3: Compare with the Given Equation

We are given that:

\[ \int \arccos(x) \, dx = x \arccos(x) + \int f(x) \, dx \]

By comparing this with the result of integration by parts, we see that:

\[ f(x) = \frac{x}{\sqrt{1 - x^2}} \]

Final Answer

The function \( f(x) \) is:

\[ f(x) = \frac{x}{\sqrt{1 - x^2}} \]