91Ó°ÊÓ

When dealing with improper integrals like the given example, determining convergence is crucial. A convergent integral is one that approaches a finite value as the limits of integration extend towards infinity, or as the point of discontinuity is approached. In the exercise, the integral \[\int_{0}^{3} x^{-1/2}(1+x) \, dx\]looks at the behavior of the integrand at the lower limit, where the function becomes undefined at \(x = 0\). To check for convergence, the integral is split into parts that are analyzed separately. If the integral of each part approaches a finite limit, the entire integral converges. Thus, the integral is convergent in this case, confirmed by finding the limit as \(x\) approaches 0 from the right (\(x \to 0^+\)).

Improper integrals can also diverge. Divergence occurs when the value of an integral does not approach a finite number when calculated over its limits. The key to identifying a divergence is in examining points where the function may not behave well, such as at infinity or a point where the function is undefined. For instance, if an integral has a term like \(x^{-1}\), it might signal divergence since it involves behavior that could lead to infinity as \(x\) approaches zero. However, in this exercise, divisional analysis revealed that both individual integrals of \(x^{-1/2}\) and \(x^{1/2}\) result in finite values. This meant that any suspected divergence due to the undefined point at \(x=0\) was resolved, indicating the integral actually converges.

A definite integral is the evaluation of the integral between two specific bounds, and provides a single value as a result. In this exercise, the focus is on integrating from 0 to 3, which makes the integral definite. It requires us to find the limits of integration applied to the antiderivative of the function. Here:

1. First, find the antiderivative for each simplified part: \(2x^{1/2}\) and \(\frac{2}{3}x^{3/2}\).
2. Second, apply the definite limits by evaluating from 0 to 3.

The result for each part is:

• For \(2x^{1/2}\), from 0 to 3, it evaluates to \(2\sqrt{3}\).
• For \(\frac{2}{3}x^{3/2}\), it evaluates to another \(2\sqrt{3}\).

Adding these gives the total definite integral value of \(4\sqrt{3}\).

Unlike definite integrals, indefinite integrals do not have bounds. They describe a family of functions and include a constant of integration, denoted as \(C\). Evaluating the indefinite integral provides the antiderivative of a function. In step 3 of the exercise, the indefinite integrals found were:

• \(\int x^{-1/2} \, dx = 2x^{1/2} + C_1\)
• \(\int x^{1/2} \, dx = \frac{2}{3}x^{3/2} + C_2\)

By finding these antiderivatives, you can then compute definite integrals by applying the boundaries. The constants \(C_1\) and \(C_2\) cancel each other out when evaluating the difference caused by bounds, leading to precise solutions in definite integrals.