Question

Evaluate \(\int_{2}^{\infty} \frac{d x}{\left(x^{2}-1\right)^{3 / 2}}\)

Short answer

The improper integral converges to \(-\frac{1}{\sqrt{3}}\).

Step-by-step solution

Identify the Type of Integral

The given integral is an improper integral because it has an infinite limit of integration. This means we can evaluate it by replacing the infinity with a variable and then taking a limit.

Replace Infinity with a Variable

Rewrite the integral, replacing the upper limit of infinity with a variable, say \(b\). The integral becomes \(\int_{2}^{b} \frac{d x}{\left(x^{2}-1\right)^{3/2}}\). This will allow us to evaluate the definite integral first, and then evaluate the limit as the variable approaches infinity.

Use Trigonometric Substitution

Consider the substitution \(x = \sec(\theta)\), as this will simplify the integral since \(x^2 - 1 = \sec^2(\theta) - 1 = \tan^2(\theta)\). With \(dx = \sec(\theta)\tan(\theta) d\theta\), the integral becomes \(\int \frac{\sec(\theta)\tan(\theta)}{(\tan^{2}(\theta))^{3/2}} d\theta\). Simplify to get \(\int \cos(\theta) d\theta\).

Determine New Bounds

When \(x=2\), \(2 = \sec(\theta)\) means \(\theta = \sec^{-1}(2)\), and when \(x=b\), \(\theta = \sec^{-1}(b)\). The integral bounds now change from \(\sec^{-1}(2)\) to \(\sec^{-1}(b)\).

Evaluate the Integral

Integrate \(\int \cos(\theta) d\theta = \sin(\theta) + C\), a typical result for the cosine function. Evaluate this from \(\theta = \sec^{-1}(2)\) to \(\theta = \sec^{-1}(b)\). This gives \(\sin(\sec^{-1}(b)) - \sin(\sec^{-1}(2))\).

Take the Limit

Taking the limit as \(b\) approaches infinity, \(\sin(\sec^{-1}(b)) = \frac{1}{\sqrt{b^2-1}}\) approaches zero, since \(b\) grows without bounds. Thus, the expression simplifies to \(- \sin(\sec^{-1}(2)) = -\frac{1}{\sqrt{4-1}} = -\frac{1}{\sqrt{3}}\).

Conclude Based on Convergence

Since the limit exists and is finite, the improper integral is convergent, and we have evaluated it successfully.

Key concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique used in calculus to simplify certain types of integrals. It is particularly useful when dealing with expressions involving square roots of quadratic terms. This method transforms a complicated expression into a simpler trigonometric one, making the integration process more straightforward.

In the given exercise, we encounter the term \(x^2-1\), which suggests using trigonometric substitution. By setting \(x = \sec(\theta)\), we leverage the identity \(\sec^2(\theta) - 1 = \tan^2(\theta)\). This transformation simplifies the radical expression into something more manageable.

• This method helps convert the integral into a form with which we are more familiar, such as \(\tan(\theta)\) or \(\sec(\theta)\).
• It uses known trigonometric identities and derivatives, making calculations easier.
• Derivatives like \(dx = \sec(\theta)\tan(\theta) d\theta\) further aid in simplifying the integration process.

After substituting, the integral simplifies to \(\int \cos(\theta) d\theta\), which is easy to solve compared to the original expression. Thus, trigonometric substitution transforms the problem into a simpler, standard form.

Convergence of Integrals

One of the key aspects of dealing with improper integrals, such as the one in this exercise, is determining whether they converge or diverge. An improper integral has at least one of its limits of integration as infinity or involves an infinite discontinuity within the limits.

To determine convergence, consider the expression of the integral and its behavior as it approaches infinity. If the limit of the integral exists and is finite as the upper bound approaches infinity, the integral converges.

• The step-by-step analysis begins by replacing infinity with a variable, \(b\), and evaluating the integral from a finite lower bound to \(b\).
• If, after evaluating, the limit as \(b \to \infty\) results in a finite value, the original integral is said to be convergent.
• In the exercise, after calculating the integral within the finite bounds and taking the limit, the resulting finite value signifies convergence.

The integral \(\int_{2}^{b} \frac{dx}{(x^2-1)^{3/2}}\) converges to \(-\frac{1}{\sqrt{3}}\), which is a clear indication of its convergent nature.

Limit Evaluation

Evaluating the limit is a crucial step in solving improper integrals, particularly when determining convergence. By taking the limit, we translate the seemingly infinite problem into a manageable calculation.

In this exercise, after substituting and evaluating the integral, we next focus on the limit of the resulting expression as \(b\) approaches infinity.

• The key point here is the behavior of \(\sin(\sec^{-1}(b)) = \frac{1}{\sqrt{b^2-1}}\), which decreases to zero as \(b\) becomes infinitely large.
• This simplification is essential, as it allows us to conclude that as the variable heads to infinity, the influence of the upper bound disappears.
• Thus, the finite negative value from the lower bound persists, leading to the final result \(-\frac{1}{\sqrt{3}}\).

By focussing on the limit evaluation, we ultimately determine that the improper integral converges and find its precise value. This meticulous examination of the limit reinforces the importance of limits in calculus.