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Expectation, also known as expected value or mean, is a key concept in probability theory. It's often denoted as \(E(X)\), representing the average outcome you would expect to see if you could repeat a random experiment infinitely many times. In simple terms, expectation is a way of summarizing what values a random variable might take on, weighted by their probabilities.

To find the expectation of a continuous random variable, you use its probability density function (PDF). The general formula for expectation when you have a PDF, \(f(x)\), is:

• \[ E(X) = \int_{a}^{b} x \, f(x) \, dx \]

Here, \(a\) and \(b\) are the bounds of integration, representing the range of the variable. This integral, in essence, adds up all possible values of \(x\) each weighted by their likelihood according to the PDF, resulting in the expected value. In our exercise, the PDF is specific and the interval is from \(0\) to \(\infty\).

A random variable is a numeric outcome of a random process or experiment. It assigns real numbers to each outcome in the sample space of a stochastic process. There are two main types of random variables: discrete and continuous.

• Discrete random variables take on a countable number of distinct values, like the roll of a die.
• Continuous random variables can take an infinite number of values. These are usually associated with measurements like height or temperature.

In our exercise, the random variable \(X\) is continuous, signified by its probability density function (PDF), \(f(x)\). The important thing about continuous random variables is that their probability is described over an interval rather than specific points, which is why we use integration to find expectations.

Integration by substitution is a vital technique used to simplify complex integrals, much like how reverse chain rule is used in differentiation. The goal is to transform a difficult integral into a simpler one by changing the variable of integration.

Let's break it down with the substitution step in our exercise:

• We set \(x = \tan(u)\). This means that \(dx = \sec^2(u) \, du\).
• This substitution also changes \(1 + x^2\) to \(\sec^2(u)\).
• The limits of integration change accordingly, making the interval from \(0\) to \(\pi/2\).

By substituting in these ways, we make the integral easier to evaluate, using familiar trigonometric integrals instead of complicated algebraic ones.

Trigonometric substitution is a powerful technique for solving integrals that involve square roots or squares, which are difficult to integrate directly. This method involves substituting a trigonometric function for a variable to simplify the integral.

In our exercise, since the problem contains \((1 + x^2)\), we use \(x = \tan(u)\), turning the integral over to a form involving \(\sec(u)\). Not only does this make the integration simpler, but it exploits the identities and properties of trigonometric functions like sine, cosine, and tangent to evaluate integrals effectively:

• We used the identity \(1 + \tan^2(u) = \sec^2(u)\).
• The substitution results in an integral of \(\sin(2u)\), which is straightforward to solve using basic sine integration techniques.

This process shows how trigonometric identities and substitutions can work together to solve otherwise challenging problems.