Question

Evaluate each improper integral or show that it diverges. \(\int_{9}^{\infty} \frac{x d x}{\sqrt{1+x^{2}}}\)

Short answer

The integral diverges.

Step-by-step solution

Identify the Type of Improper Integral

The integral \( \int_{9}^{\infty} \frac{x \, dx}{\sqrt{1+x^2}} \) has an upper limit of infinity, making it an improper integral of type I. This requires us to evaluate the limit as the upper bound goes to infinity.

Set Up the Limit for the Improper Integral

Rewrite the improper integral as a limit: \[ \int_{9}^{\infty} \frac{x \, dx}{\sqrt{1+x^2}} = \lim_{b \to \infty} \int_{9}^{b} \frac{x \, dx}{\sqrt{1+x^2}} \]

Perform a Substitution

Use the substitution \( u = 1 + x^2 \), which implies \( du = 2x \, dx \) or \( x \, dx = \frac{1}{2} du \). The limits of integration change: when \( x = 9 \), \( u = 82 \); and when \( x = b \), \( u = 1 + b^2 \).

Change of Variables in the Integral

Substitute into the integral:\[ \lim_{b \to \infty} \int_{82}^{1+b^2} \frac{1}{2} \frac{1}{\sqrt{u}} \, du \]This simplifies to:\[ \frac{1}{2} \lim_{b \to \infty} \int_{82}^{1+b^2} u^{-1/2} \, du \]

Integrate the Simplified Integral

The integral of \( u^{-1/2} \) is \( 2u^{1/2} \). Calculate the integral:\[ \frac{1}{2} \cdot 2 \left[ u^{1/2} \right]_{82}^{1+b^2} = \lim_{b \to \infty} \left[ u^{1/2} \right]_{82}^{1+b^2} \]

Evaluate the Limit

Evaluate the expression:\[ \lim_{b \to \infty} \left[ (1+b^2)^{1/2} - 82^{1/2} \right] = \lim_{b \to \infty} (\sqrt{1+b^2} - \sqrt{82}) \] As \( b \to \infty \), \( \sqrt{1+b^2} \to b \), so the limit is infinite. The integral diverges.

Key concepts

Understanding Divergence in Improper Integrals

When tackling the concept of divergence in improper integrals, it's essential to know when an integral does not converge to a finite value. In our exercise, we evaluated \( \int_{9}^{\infty} \frac{x \, dx}{\sqrt{1+x^{2}}} \). This specific improper integral diverges. But what does divergence mean?
Divergence occurs when the value of the integral becomes infinitely large as the bound approaches its limit. For example, if the result of our evaluated integral heads towards infinity as we extend the upper bound, we say the integral diverges. Most often, divergent integrals involve cases where either the function itself becomes unbounded or the limits are infinite. Understanding this is crucial since it helps in identifying integrals that don't settle to a neat, finite number.

In summary:

• Divergence means the integral doesn't resolve to a bounded value.
• This typically happens with infinite boundaries or unbounded functions.

Recognizing divergence early helps in anticipating infinite results or suggesting alternative strategies.

Exploring Limit Evaluation in Improper Integrals

Limit evaluation is the key technique for dealing with improper integrals. When you have an integral like \( \int_{9}^{\infty} \frac{x \, dx}{\sqrt{1+x^{2}}} \), you're faced with an upper bound of infinity. To handle this infinity, we replace the integral expression with a limit involving a variable approaching infinity.
The original problem is rewritten as a limit problem: \[ \lim_{b \to \infty} \int_{9}^{b} \frac{x \, dx}{\sqrt{1+x^2}} \]
This transformation is crucial because it allows you to proceed with conventional calculus techniques, such as substitution, knowing that you'll ultimately need to plug infinity into your resulting expression.

Here’s why limit evaluation is important:

• It allows you to deal with infinite bounds more meaningfully.
• Once in limit form, you can use different methods to simplify and solve.

Evaluating limits is tough, but with practice, it becomes a powerful tool for understanding behavior at infinity.

The Substitution Method in Calculating Integrals

The substitution method is a fantastic way to simplify complex integrals. In our exercise, we used the substitution \( u = 1 + x^2 \). This changes both the limits of integration and the differential, transforming the integral into a more manageable form.
The choice of substitution depends largely on identifying parts of the integrand that can be neatly expressed in terms of a single new variable, often simplifying the integrand or accommodating the differential's nuances.

With substitution:

• The integral becomes more straightforward with respect to the new variable.
• The bounds change according to this substitution, aligning with the new variable's terms.
• Calculations post-substitution must respect the new variable's impact on limits.

This method is predictive and intuitive once you catch onto patterns, providing powerful insight into integral evaluation.

Dealing with Infinite Bounds in Integrals

Infinite bounds introduce a unique challenge in integral calculus. Integrals like \( \int_{9}^{\infty} \frac{x \, dx}{\sqrt{1+x^2}} \) exemplify situations where you encounter infinity as a limit.
Mathematically evaluating such integrals involves understanding how the function behaves as it stretches towards infinity. In practice, applying a limit lets us say, 'What if we went further?' specifically \( \lim_{b \to \infty} \int_{9}^{b} \cdots \). This helps us analyze and grasp what the integral might do—including divergence, where it heads to infinity instead of any finite conclusion.

Managing infinite bounds includes:

• Converting the problem into a limit problem.
• Tracking changes in variable and notation to cater to infinity's influence.
• Recognizing and testing when functions remain bounded or otherwise.

Working with infinite bounds lets mathematicians predict the behavior of curves and areas within immense spans.