Problem 30 - Calculus for Biology and Medicine

Question

Determine whether each integral is convergent. If the integral is convergent, compute its value. $$ \int_{-\infty}^{1} \frac{3}{1+x^{2}} d x $$

Step-by-Step Solution

Answer

The integral is convergent and its value is \( \frac{9\pi}{4} \).

Step 1: Recognize the Type of Integral

This is an improper integral because the lower limit of integration is \(\-\infty\). It needs to be evaluated as a limit because it involves an infinite limit of integration.

Step 2: Set Up the Limit

Replace the lower limit \(\-\infty\) with a variable \(a\) and take the limit as \(a\) approaches \(\-\infty\). The integral becomes \( \int_{a}^{1} \frac{3}{1+x^{2}}\, dx \)

Step 3: Perform the Integration

The integral of \(\frac{3}{1+x^2}\) is a standard form. The antiderivative is \(\3\arctan(x)\). So, the integral becomes $\[\3\left[ \arctan(x) \right]_{a}^{1}\].

Step 4: Evaluate the Definite Integral

Substitute the limits into the antiderivative: \[\3\left(\arctan(1) - \arctan(a)\right)\]. Since \(\\arctan(1) = \frac{\pi}{4}\) and \(\\arctan(a)\) approaches \(\-\frac{\pi}{2}\) as \(\a \to -\infty\), substitute these into the expression.

Step 5: Compute the Limit

Evaluate as \(\ a \to -\infty\): \[\ 3\left(\frac{\pi}{4} - \left(-\frac{\pi}{2}\right) \right) \] simplifies to \[\ 3\left(\frac{\pi}{4} + \frac{\pi}{2}\right) \ = \frac{3\pi}{4} + \frac{3\pi}{2}\].

Step 6: Simplify the Result

Combine the terms: \[\frac{3\pi}{4} + \frac{6\pi}{4} = \frac{9\pi}{4}.\]

Key Concepts

Convergence of Integrals

When we are dealing with improper integrals, it is crucial to determine whether the integral is convergent or divergent. An integral is convergent if it approaches a finite value as the limit of integration moves towards infinity. Conversely, an integral is divergent if it does not settle at a finite number, drifting towards infinity or oscillating indefinitely.
A common scenario with improper integrals involves functions defined over an infinite interval, like in our original exercise: \\(\int_{-\infty}^{1} \frac{3}{1+x^{2}} dx\). Here, the integration starts from \(-\infty\), making it an improper integral.
To investigate convergence, we introduced a variable, say \(a\), replacing \(-\infty\). The limit \( \lim_{a \to -\infty} \) of the definite integral from \(a\) to 1 is used to determine convergence. By evaluating \\(\lim_{a \to -\infty} \int_{a}^{1} \frac{3}{1+x^{2}} dx\), if we find a finite value, like \(\frac{9\pi}{4}\) in our solution, the integral is convergent. This means the area under the curve, as calculated over this infinite interval, is bounded and precise.

Integration Techniques

Various integration techniques help us tackle improper integrals. Identify the correct one based on the function form. For functions like \(\frac{3}{1+x^2}\), recognize it as a standard form for integration. The antiderivative is straightforward, and in our specific case, it results in \(3\arctan(x)\).
Integrating step-by-step:

• First, replace \(-\infty\) in the integral with a variable, say \(a\).
• Write the new integral: \(\int_{a}^{1} \frac{3}{1+x^2}\, dx\).
• Find the antiderivative of the integrand, here it is \(3\arctan(x)\).
• Compute the definite integral by substituting the limits into the antiderivative.

Once the definite integral is evaluated, switch back to using limits to solve improper integrals effectively. Each form of integrals demands different tricks; thus, familiarizing oneself with these techniques aids essential calculus skills.

Limits and Antiderivatives

Limits are fundamental when dealing with improper integrals. They gauge how the function behaves as it approaches extreme bounds, like infinity. For the integral \(\int_{-\infty}^{1} \frac{3}{1+x^2} dx\), we needed to define the antiderivative first, which was \(3\arctan(x)\).
The process involved substituting the limits into the antiderivative function. Start with evaluating the

• The antiderivative here is determined to be \(3\arctan(x)\).
• Next, substitute the upper bound 1 to get \(3\arctan(1)\).
• And then, substitute the variable \(a\) representing \(-\infty\) for the lower bound to get \(3\arctan(a)\).

Understanding these steps ensures you get accurate estimates of the area under curves. When \(a\) approaches \(-\infty\), observe \(\arctan(a)\) approaches \(-\frac{\pi}{2}\). These are combined under the limit: \\(3\left(\frac{\pi}{4} + \frac{\pi}{2}\right)\), yielding a value of \(\frac{9\pi}{4}\). With this methodology, harnessing limits is powerful in integrating functions over infinite ranges.