Random Variable with Density Function
The random variable x has a density function given by f(x) = c/(x^2 + 1). Here, c is a constant that ensures the total area under the density curve is equal to 1. This density function represents a continuous probability distribution for the random variable x.

Density Function f(x)
The density function f(x) = c/(x^2 + 1) is defined for all real numbers x. The function is non-negative for all x, as the square of x and 1 are both positive. This ensures that the density function is valid and can be used to calculate probabilities.

Calculating the Constant c
To find the constant c, we need to ensure that the total area under the density curve is equal to 1. This can be done by integrating the density function over the entire range of x and setting it equal to 1. The integral of f(x) over all x should be equal to 1 to satisfy the property of a probability density function.

Interpreting the Density Function
The density function f(x) = c/(x^2 + 1) describes the likelihood of the random variable x taking on different values. As x increases, the density decreases, but it never reaches zero. This indicates that x can take on any real value with a certain probability as defined by the density function.
Using this density function, we can calculate probabilities associated with different intervals of x and make inferences about the random variable x in question.