Problem 12 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated integrations. \(\int_{2}^{3} \frac{d t}{t^{2} \sqrt{t^{2}-1}}\)

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{4\sqrt{2} - 3\sqrt{3}}{6}\).

Step 1: Identify Substitution for Integration

The integrand is in the form \[ \frac{1}{t^2 \sqrt{t^2 - 1}} \]This suggests that a trigonometric substitution might simplify the integral. For this form, let us use\[ t = \sec(\theta) \]which implies that \[ dt = \sec(\theta) \tan(\theta) d\theta. \]

Step 2: Change Limits of Integration

Convert the limits of integration from terms of \(t\) to \(\theta\). When \(t = 2\), then \[ 2 = \sec(\theta) \Rightarrow \theta = \cos^{-1}\left(\frac{1}{2}\right). \]For \(t = 3\),\[ 3 = \sec(\theta) \Rightarrow \theta = \cos^{-1}\left(\frac{1}{3}\right). \]

Step 3: Substitute and Simplify the Integral

By substitution, the integral becomes\[ \int_{\cos^{-1}\left(\frac{1}{2}\right)}^{\cos^{-1}\left(\frac{1}{3}\right)} \frac{\sec(\theta) \tan(\theta)}{\sec^2(\theta) \cdot \tan(\theta)} d\theta \]simplifying to\[ \int_{\cos^{-1}\left(\frac{1}{2}\right)}^{\cos^{-1}\left(\frac{1}{3}\right)} \frac{1}{\sec(\theta)} d\theta = \int_{\cos^{-1}\left(\frac{1}{2}\right)}^{\cos^{-1}\left(\frac{1}{3}\right)} \cos(\theta) d\theta. \]

Step 4: Integrate and Evaluate

The integral of \(\cos(\theta)\) is \(\sin(\theta)\). Evaluating this from \(\cos^{-1}\left(\frac{1}{2}\right)\) to \(\cos^{-1}\left(\frac{1}{3}\right)\), we get\[ \sin\left(\cos^{-1}\left(\frac{1}{3}\right)\right) - \sin\left(\cos^{-1}\left(\frac{1}{2}\right)\right). \]Using the identity \( \sin(\cos^{-1}(x)) = \sqrt{1-x^2} \), compute:\[ \sin\left(\cos^{-1}\left(\frac{1}{3}\right)\right) = \sqrt{1-\left(\frac{1}{3}\right)^2} = \sqrt{1 - \frac{1}{9}} = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3}, \]\[ \sin\left(\cos^{-1}\left(\frac{1}{2}\right)\right) = \sqrt{1-\left(\frac{1}{2}\right)^2} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}. \]

Step 5: Calculate the Result

Substitute back to find the final result:\[ \frac{2\sqrt{2}}{3} - \frac{\sqrt{3}}{2} = \frac{4\sqrt{2} - 3\sqrt{3}}{6}. \]Compute this expression to approximate the numerical value if necessary.

Key Concepts

Definite Integration

Definite integration involves finding the integral of a function over a specific interval. This process calculates the net area under the curve of the function from one point to another. It's important to note that in definite integration:

• The limits of integration are specified, typically as a lower and an upper bound.
• The result is a number, unlike indefinite integration which results in a function.
• One uses the Fundamental Theorem of Calculus to evaluate the integral at these bounds.

In the original exercise, we evaluate the definite integral of \(\int_{2}^{3} \frac{d t}{t^{2} \sqrt{t^{2}-1}}\)between the limits 2 and 3. The aim is to compute the total area under the curve \( \frac{1}{t^{2} \sqrt{t^{2}-1}} \)from \( t = 2 \)to \( t = 3. \)"Definite" means we're interested in this specific interval, and the integration will give us a concrete value.

Integration by Substitution

Integration by substitution is a crucial method used to simplify complex integrals by changing variables. This technique often transforms the integral into a more manageable form. Here are the steps involved:

• Identify the substitution: This involves finding a trigonometric or algebraic function that can simplify the integral.
• Change variables: Substitute the chosen function and its differential into the integral.
• Back-substitute: Once integrated, revert to the original variable if necessary.

In our exercise, we used trigonometric substitution with \( t = \sec(\theta) \)because it directly addressed the \( \sqrt{t^2 - 1} \)term in the integrand. This simplification allowed us to rewrite the integral in terms of \( \theta \), facilitating easier evaluation. Moreover, the limits of integration were adjusted from \( t \) terms to \( \theta \) terms, ensuring correct bounds after substitution.

Integration Techniques

The process of integration often involves choosing from various techniques to make calculations manageable. Key techniques include:

• Trigonometric Substitution: Used when encountering square roots of quadratic expressions (e.g., \( \sqrt{t^2 - 1} \)). This can simplify integrals into basic trigonometric forms.
• Partial Fraction Decomposition: Useful for rational functions. It involves breaking down complex fractions into simpler parts.
• Integration by Parts: Helpful for products of functions, using a formula akin to the product rule for differentiation.

For the given integral, trigonometric substitution was not just an option but a necessity. This method was ideal because the expression \( \sqrt{t^2 - 1} \)fits the perfect trigonometric identity \( \sec^2(\theta) - 1 = \tan^2(\theta) \).Identifying the right technique is essential for simplifying the integral and achieving an accurate result efficiently.