Use the probability density function given below to find the indicated probabilities. Illustrate each probability with a graph. \[ f(x)=\left\{\begin{array}{ll} \frac{1}{(1+x)^{2}} & \text { if } x \geq 0 \\ 0 & \text { otherwise } \end{array}\right. \] (A) $P(1 \leq X \leq 5)$ (B) $P(X>2)$ (C) $P(X \leq 5)$ (A) $P(1 \leq X \leq 5)=\square$ (Type an integer or a simplified fraction.)

Step 1 ：The question is asking for the probability that a random variable X, which follows the given probability density function, falls within certain ranges. The probability that a continuous random variable falls within a certain range is given by the integral of the probability density function over that range.

Step 2 ：For part (A), we need to find the probability that X is between 1 and 5. This is given by the integral from 1 to 5 of the function \(f(x)\).

Step 3 ：The result of the integral is approximately 0.333, which represents the probability that the random variable X falls between 1 and 5.

Step 4 ：Final Answer: \(P(1 \leq X \leq 5)=\boxed{\frac{1}{3}}\)