Problem 27 - Calculus Single Variable

Question

Evaluate the given integral by making a trigonometric substitution (even if you spot another way to evaluate the integral). $$ \int_{\sqrt{5}}^{\sqrt{10}} \frac{6}{\left(x^{2}-1\right)^{3 / 2}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( 3\sqrt{5} - 2\sqrt{10} \).

Step 1: Identify the Trigonometric Substitution

Observe that the integral has the form \( \int \frac{6}{(x^2 - 1)^{3/2}} \, dx \). This suggests the use of a trigonometric substitution such as \( x = \sec(\theta) \), because \( \sec^2(\theta) - 1 = \tan^2(\theta) \).

Step 2: Perform the Substitution

With the substitution \( x = \sec(\theta) \), differentiate to get \( dx = \sec(\theta) \tan(\theta) \, d\theta \). Substitute these into the integral to obtain: \[ \int \frac{6}{(\sec^2(\theta) - 1)^{3/2}} \cdot \sec(\theta) \tan(\theta) \, d\theta = \int \frac{6 \sec(\theta) \tan(\theta)}{\tan^3(\theta)} \, d\theta. \] Notice that \( (\sec^2(\theta) - 1)^{3/2} = \tan^3(\theta) \), simplifying the integral.

Step 3: Simplify the Integral

The integral simplifies to \[ \int \frac{6 \sec(\theta) \tan(\theta)}{\tan^3(\theta)} \, d\theta = \int \frac{6 \sec(\theta)}{\tan^2(\theta)} \, d\theta = \int \frac{6 \cot(\theta)}{\sin(\theta)} \, d\theta. \]

Step 4: Use Identity and Integral Formula

Use the identity \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \) to rewrite the integral as \[ \int 6 \cot(\theta) \csc(\theta) \, d\theta = \int 6 \frac{\cos(\theta)}{\sin^2(\theta)} \, d\theta. \]Recognize this as the derivative of \( \csc(\theta) \), leading to \[ -6 \csc(\theta) + C. \]

Step 5: Back-Substitute

Since \( x = \sec(\theta) \), it follows that \( \theta = \sec^{-1}(x) \), and so \( \sin(\theta) = \frac{\sqrt{x^2 - 1}}{x} \) and \( \csc(\theta) = \frac{x}{\sqrt{x^2 - 1}} \). Thus, the integral becomes \[ -6 \cdot \frac{x}{\sqrt{x^2 - 1}}. \]

Step 6: Evaluate the Definite Integral

Evaluate the definite integral from \( x = \sqrt{5} \) to \( x = \sqrt{10} \). Calculate the antiderivative at these points:\[ -6 \cdot \frac{\sqrt{10}}{\sqrt{(\sqrt{10})^2 - 1}} \Bigg|^\sqrt{10}_\sqrt{5} = -6 \cdot \left(\frac{\sqrt{10}}{3}\right) + 6 \cdot \left(\frac{\sqrt{5}}{2}\right). \] Simplify this to get:\[ -6 \cdot \frac{\sqrt{10}}{3} + 6 \cdot \frac{\sqrt{5}}{2} = -2\sqrt{10} + 3\sqrt{5}. \]

Step 7: Final Simplification

The final simplified value of the evaluated integral is \( 3\sqrt{5} - 2\sqrt{10} \).

Key Concepts

Definite Integral

A definite integral is a crucial concept in calculus, representing the area under a curve between two specified points on the x-axis. In this exercise, we deal with calculating the definite integral of a function using trigonometric substitution. This entails evaluating the integral within specific boundaries, in this case from \( x = \sqrt{5} \) to \( x = \sqrt{10} \).

To solve the integral \( \int_{\sqrt{5}}^{\sqrt{10}} \frac{6}{(x^2 - 1)^{3/2}} \, dx \), we transform it via substitution into an easier form that can be integrated. The definite integral then requires us to find the antiderivative of the function and evaluate it at the upper and lower limits. This process results in values that provide the net area under the curve, considering the subtraction of the lower limit evaluation from the upper limit one.

• The integral calculates accumulated change or area.
• Boundary values are vital for evaluating the integral.
• Commonly used in physics and engineering for calculating quantities like displacement or total work done.

Understanding how to manipulate these components with different techniques, such as trigonometric substitution, enriches your toolkit in solving more complex integral calculus problems.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domain. These identities are key to simplifying integrals in calculus, such as when dealing with expressions like those found in this exercise.

In the example provided, we used the identity \( \sec^2(\theta) - 1 = \tan^2(\theta) \) to transform the original expression. The choice of the substitution \( x = \sec(\theta) \) was particularly useful because it exploits this identity, allowing the integral to be rewritten in terms of \( \theta \) rather than \( x \), greatly simplifying the process.

• Helps in transforming and simplifying equations.
• Reduces complicated expressions into more manageable forms.
• Critical for solving integrals that are not immediately straightforward.

By understanding and correctly applying trigonometric identities, you are able to tackle integrals that would otherwise be too cumbersome to evaluate using basic algebraic methods alone.

Integral Calculus

Integral calculus focuses on the mathematical study of integrals and their properties. It revolves around the accumulation of quantities, such as areas under a curve, as we analyze in this exercise. When calculating integrals, we aim to determine functions that describe how quantities change cumulatively.

The trigonometric substitution used here is a classical technique in integral calculus that facilitates the computation by converting the original integral into a simpler form. This technique leverages trigonometric identities to manage expressions that involve roots and their powers, common in physical-based problems and theoretical mathematics.

• Explores the concept of antiderivatives.
• Used in calculating areas, volumes, and total change.
• Integrates several nuances of algebra and trigonometry.

Mastery of integral calculus often requires a blend of creativity and technical skill, as you learn to recognize the underlying patterns that allow these transformations. As you become more adept, you'll find that you can approach a wide range of problems with confidence and precision.