Problem 20 - Calculus Concepts

Question

For Activities 17 through \(22,\) use algebraic manipulation or integration by substitution as well as limits to evaluate the improper integral. $$ \int_{2}^{\infty} x^{3} \sqrt{x^{4}+1} d x $$

Step-by-Step Solution

Answer

The integral diverges and approaches infinity.

Step 1: Define the Improper Integral and Apply a Limit

Identify the integral as improper because the upper bound is infinity. To evaluate it, rewrite the integral as a limit: \[ \int_{2}^{\infty} x^{3} \sqrt{x^{4}+1} \, dx = \lim_{b \to \infty} \int_{2}^{b} x^{3} \sqrt{x^{4}+1} \, dx. \]

Step 2: Analyze for Substitution

Consider a substitution that simplifies the problem. Let \( u = x^4 + 1 \), which makes \( du = 4x^3 \, dx \) or \( x^3 \, dx = \frac{1}{4} du \).

Step 3: Change the Limits of Integration

When \( x = 2 \), \( u = 2^4 + 1 = 17 \). As \( x = b \), \( u = b^4 + 1 \). The integral becomes:\[ \lim_{b \to \infty} \int_{17}^{b^4 + 1} \sqrt{u} \cdot \frac{1}{4} \, du. \]

Step 4: Evaluate the Integral with Respect to \( u \)

The integral now is:\[ \frac{1}{4} \int \sqrt{u} \, du = \frac{1}{4} \cdot \frac{2}{3} u^{3/2} = \frac{1}{6} u^{3/2}. \]

Step 5: Substitute Back and Evaluate

Evaluate \( \frac{1}{6} u^{3/2} \) from 17 to \( b^4 + 1 \). Substitute the limits:\[ \lim_{b \to \infty} \left[ \frac{1}{6} (b^4 + 1)^{3/2} - \frac{1}{6} \, 17^{3/2} \right]. \]

Step 6: Evaluate the Limit

Analyze the limit as \( b \to \infty \). The term \( (b^4 + 1)^{3/2} \) dominates and approaches infinity. Thus, the expression diverges.\[ \lim_{b \to \infty} \frac{1}{6} (b^4 + 1)^{3/2} = \infty. \]

Key Concepts

Integration by Substitution

Integration by substitution is a powerful technique allowing us to simplify complicated integrals. By replacing complex expressions with simpler variables, integrals become much easier to solve. This is especially helpful when dealing with functions that appear combined, such as products or quotients.

In this problem, we identified an expression that would simplify our integral: if we let \( u = x^4 + 1 \), then the differential \( du = 4x^3 \, dx \) can replace the \( x^3 \, dx \) term in our integrand when adjusted by the constant factor of \( \frac{1}{4} \). This substitution transforms a cumbersome integral into a simpler one involving \( \sqrt{u} \), which is much easier to integrate.

• Select a part of the integrand that simplifies when substituted.
• Calculate the derivative to find \( du \) in terms of \( dx \).
• Adjust your substitution to match the original integral's form.

With practice, the process of identifying suitable substitutions becomes intuitive, streamlining the integration process dramatically.

Limits in Calculus

Limits in calculus are fundamental to dealing with improper integrals, as they allow us to define what happens as certain variables approach infinity. An improper integral has one or both limits of integration as infinity or involves an integrand that approaches infinity within the limits of integration.

In our example, we tackle the improper integral \( \int_{2}^{\infty} x^3 \sqrt{x^4 + 1} \, dx \) by rewriting it using a limit. This is done by introducing a parameter \( b \) that approaches infinity:

• Set up the integral with a limit such as \( \lim_{b \to \infty} \int_{2}^{b} \ldots \, dx \).
• Evaluate the integral with respect to the finite limit \( b \), and then analyze how the result behaves as \( b \to \infty \).
• This approach allows us to determine whether the integral converges to a finite value or diverges as an infinite limit results.

Thus, the use of limits is essential for the evaluation of such integrals, providing a logical framework for understanding infinite domains.

Divergence of Integrals

Divergence of integrals is an important concept that tells us whether an integral does not settle to a finite value. When exploring the behavior of improper integrals, like \( \int_{2}^{\infty} x^3 \sqrt{x^4 + 1} \, dx \), we encounter this issue.

In the solved example, after substituting and integrating the simplified expression, the final step involves evaluating the limit:

• As the variable \( b \to \infty \), evaluate the expression \( \frac{1}{6} (b^4 + 1)^{3/2} \).
• The expression grows without bound, indicating that it does not converge to a finite number.
• Thus, the integral is said to diverge because it approaches infinity rather than a specific value.

Divergence typically flags that the area under the curve represented by the integral infinitely expands, and so it does not make sense to assign a finite numerical result. Exploring divergence helps us understand the limitations of integration and the behavior of functions across unbounded intervals.