Problem 27 - Calculus Single Variable

Question

Determine whether the given improper integral converges or diverges. If it converges, then evaluate it. $$ \int_{-\infty}^{0} \frac{1}{\left(1+x^{2}\right)^{3 / 2}} d x $$

Step-by-Step Solution

Answer

The integral converges and its value is 1.

Step 1: Identify the Problem Type

This is an improper integral because it has an infinite limit of integration (from \(-\infty\) to 0). We need to determine if it converges (i.e., has a finite value) or diverges (i.e., does not have a finite value).

Step 2: Use Substitution for Integration

To simplify calculation, we perform substitution. Let \( x = \tan \theta \), which implies \( dx = \sec^2 \theta \, d\theta \). Note that the denominator becomes \((1+x^2)^{3/2} = (\sec^2 \theta)^{3/2} = \sec^3 \theta\). Change the limits of integration accordingly to \(\theta\) as \(x = -\infty\) corresponds to \(\theta = -\frac{\pi}{2}\) and \(x = 0\) corresponds to \(\theta = 0\).

Step 3: Set Up the Improper Integral with New Variable

Change the variable in the integral:\[\int_{-\frac{\pi}{2}}^0 \frac{\sec^2 \theta}{\sec^3 \theta} d\theta = \int_{-\frac{\pi}{2}}^0 \cos \theta d\theta\]

Step 4: Evaluate the Integral

The integral of \(\cos \theta\) is \(\sin \theta\). Evaluate it from \(\theta = -\frac{\pi}{2}\) to \(\theta = 0\):\[\left. \sin \theta \right|_{-\frac{\pi}{2}}^0 = \sin(0) - \sin\left(-\frac{\pi}{2}\right) = 0 + 1 = 1\]

Step 5: Conclusion About Convergence

Since the integral evaluates to a finite value (1), the improper integral is convergent.

Key Concepts

Convergence

The concept of convergence in the context of improper integrals is crucial for determining whether an integral has a finite value. An improper integral involves limits that are infinite or functions that approach infinity within the interval of integration.

Convergence occurs when an integral evaluates to a finite number. In that case, the integral "converges". On the contrary, if the integral does not settle to a finite value, it is said to "diverge".

• Identify whether the integral has an infinite limit of integration or if the function itself becomes infinite.
• Attempt the evaluation: if you reach a finite result, the integral converges.
• Be aware: Convergence can depend on changing variables or limits for evaluation simplicity.

Understanding how convergence is determined helps us conclude that an integral represents a well-defined area or accumulation of values, which has direct applications in physics and engineering.

Integration by Substitution

Integration by substitution is a technique used to simplify the process of finding antiderivatives, particularly useful when working with complex algebraic expressions.

This method involves changing variables via substitution, simplifying the integral into a more recognizable form.

• Choose a substitution that simplifies the expression, such as replacing parts of the function with trigonometric identities.
• Modify the limits of integration according to the new variable, providing clarity for the new bounds.
• Perform the integration in the simpler variable, then substitute back if necessary to obtain the final result.

For example, in the original problem, the substitution of \(x = \tan \theta\) significantly simplifies integrating the function relative to \(\theta\). Thus, integration by substitution is more than just a tool; it's an approach that can transform challenging problems into manageable tasks.

Trigonometric Substitution

Trigonometric substitution is a special type of substitution technique that uses trigonometric identities to simplify integrals involving square roots.

In situations where polynomials and expressions involving square roots appear, substituting using trigonometric identities can change the integral into a simpler form.

• Common substitutions depend on the form of the expression: \( x = \tan \theta \), \( x = \sin \theta \), or \( x = \cos \theta \).
• These substitutions can transform the problem into an integral in terms of \(\theta\), often reducing complex fractions or roots.
• The integrals typically result in a trigonometric function that is easier to evaluate.

Utilizing trigonometric substitution can turn complex algebraic expressions into straightforward trigonometric integrals, as we saw with the transformation into terms of \(\cos \theta\) for easier evaluation.

Infinite Limits of Integration

When dealing with improper integrals, infinite limits of integration play a key role. An integral is considered improper if it includes limits that extend to infinity.

These limits are typically from an infinite point or towards infinity. Understanding how to handle them is essential for solving these types of integrals.

• Verify whether the limits extend to infinity, indicating your integral is improper.
• Use techniques like substitution to reframe the integral in terms of finite limits, simplifying the process.
• Evaluate the resulting integral to check if it converges, determining if a finite area under the curve exists.

By managing infinite limits correctly, we ensure that the methods applied to solve improper integrals like extending limits from \(-\infty\) to 0 lead to meaningful results, as demonstrated in the example integrating \(\cos \theta\) over adjusted limits.