Problem 26 - Calculus Single Variable

Question

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int_{\sqrt{2}}^{\sqrt{10}} \frac{x^{3}}{\sqrt{x^{2}-1}} d x $$

Step-by-Step Solution

Answer

Use trigonometric substitution, evaluate via \(\sec\) and find definite integral value.

Step 1: Identify the Trigonometric Substitution

To evaluate the integral \( \int_{\sqrt{2}}^{\sqrt{10}} \frac{x^{3}}{\sqrt{x^{2}-1}} dx \), we notice that the expression inside the square root resembles \( x^2 - a^2 \) with \( a = 1 \). The standard substitution for \( x^2 - a^2 \) is \( x = a \sec(\theta) \). Thus, set \( x = \sec(\theta) \).

Step 2: Differentiate the Substitution

Differentiate \( x = \sec(\theta) \) with respect to \( \theta \) to find \( dx \):\[ dx = \sec(\theta) \tan(\theta) \, d\theta. \]

Step 3: Substitute and Simplify the Integral

Substituting \( x = \sec(\theta) \) and \( dx \), we get:\[ \int \frac{(\sec(\theta))^3}{\sqrt{(\sec^2(\theta)) - 1}} \cdot \sec(\theta) \tan(\theta) \, d\theta. \]Simplify the integrand using the identity \( \sec^2(\theta) - 1 = \tan^2(\theta) \):\[ \int \frac{\sec^4(\theta) \tan(\theta)}{\tan(\theta)} \, d\theta = \int \sec^4(\theta) \, d\theta. \]

Step 4: Find the New Limits of Integration

Calculate the new limits for \( \theta \) by considering the original limits:- When \( x = \sqrt{2} \), \( \sqrt{2} = \sec(\theta_1) \Rightarrow \theta_1 = \sec^{-1}(\sqrt{2}) \).- When \( x = \sqrt{10} \), \( \sqrt{10} = \sec(\theta_2) \Rightarrow \theta_2 = \sec^{-1}(\sqrt{10}) \).

Step 5: Integrate \( \sec^4(\theta) \, d\theta \)

The integral \( \int \sec^4(\theta) \, d\theta \) can be evaluated using the reduction formula or integration by parts. The result is:\[ \frac{1}{3} (2\tan^3(\theta) + 3\tan(\theta)) + C. \]

Step 6: Solve the Definite Integral

Evaluate the integral from \( \theta_1 \) to \( \theta_2 \):\[ \left[ \frac{1}{3} (2\tan^3(\theta) + 3\tan(\theta)) \right]_{\theta_1}^{\theta_2}. \]Use the values of \( \theta_1 \) and \( \theta_2 \) obtained earlier to compute this expression.

Step 7: Calculate the Value

Finally, substitute \( \theta_1 = \sec^{-1}(\sqrt{2}) \) and \( \theta_2 = \sec^{-1}(\sqrt{10}) \) into the expression and evaluate the definite integral to get the final numerical value.

Key Concepts

Integration Techniques

Integration techniques are essential tools for solving different types of integrals. There are several methods you can choose from, depending on the form of the function you want to integrate. One of the most important techniques is **substitution**, which is often used with trigonometric functions. Rest assured, once you master these techniques, integrals will become much more approachable.

In the original exercise, we're dealing with a function that includes a radical, specifically \( \sqrt{x^2 - 1} \). This strongly suggests using a technique known as **trigonometric substitution**. Trigonometric substitution helps simplify the integrand by replacing variable expressions with trigonometric identities.

For example, when who see an integral involving \( \sqrt{x^2 - a^2} \), try substituting \( x = a \sec(\theta) \). This allows the radical to simplify into a basic trigonometric identity because of the relationship \( \sec^2(\theta) - 1 = \tan^2(\theta) \). By simplifying the expression under the square root, substituting makes the integral easier to solve.

• Key method: Trigonometric Substitution.
• Purpose: Simplify integrals by using trigonometric identities.
• Application: Particularly useful for integrals involving \( \sqrt{x^2 \pm a^2} \).

By mastering these techniques, you’ll be able to tackle a wide range of integral problems more effectively and accurately.

Definite Integral

A definite integral differs from an indefinite integral by the presence of its limits of integration. When evaluating a definite integral, you calculate the "net area" under the curve between two specified points on the x-axis.

When we solve for definite integrals, such as in the original exercise from \( x = \sqrt{2} \) to \( x = \sqrt{10} \), we're employing a method to find the actual value associated with these bounds. Using trigonometric substitution changes the variable of integration, which means we need to update the limits of integration accordingly. In the exercise, these new limits are found by determining \( \theta_1 \) and \( \theta_2 \) using the function \( x = \sec(\theta) \), leading to \( \theta_1 = \sec^{-1}(\sqrt{2}) \) and \( \theta_2 = \sec^{-1}(\sqrt{10}) \).

• A definite integral reflects the total change, net area, or accumulated motion within the bounds.
• Converting limits during substitution ensures the evaluated integral remains correct.
• Final evaluation involves calculating the difference between two values of the antiderivative at these bounds.

Being clear on these concepts of limits and using substitution will allow the determination of definite integrals to be systematic and precise.

Trigonometric Identities

Trigonometric identities are fundamental in calculus, especially for simplifying expressions containing square roots and powers. They're useful in various methods, including substitution.

In problems that involve trigonometric substitution, these identities simplify expressions substantially. In the original exercise, the critical identity employed is: \( \sec^2(\theta) - 1 = \tan^2(\theta) \). This identity allowed the reduction of a potential complex radical expression into a power function that is much simpler to integrate.

Additionally, solving the integral \( \int \sec^4(\theta) \, d\theta \) often requires the use of trigonometric identities. The secant functions can be related back to simpler trigonometric functions by using identities such as \( \sec(\theta) = \frac{1}{\cos(\theta)} \). These are crucial because they determine how the function breaks down during integration.

• Ease of use of identities helps simplify and manage complex functions.
• Variety of identities are available, such as Pythagorean, angle sum, and double angle identities.
• Applying identities thoughtfully enables more straightforward solutions to integrals.

Mastering these identities equips you with the necessary intuition and flexibility to transform complex expressions into simpler forms and find solutions with confidence.