Problem 44 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems 1-54, perform the indicated integrations. \(\int \frac{d t}{2 t \sqrt{4 t^{2}-1}}\)

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{1}{4} \sec^{-1}(2t) + C\).

Step 1: Simplify the Integral

The given integral is \( \int \frac{d t}{2 t \sqrt{4 t^{2}-1}} \). We can take the constant \( \frac{1}{2} \) out of the integral: \( \frac{1}{2} \int \frac{d t}{t \sqrt{4t^2 - 1}} \).

Step 2: Substitution Method

To simplify the integral, let's use a substitution. Let \( u = 2t \), hence \( du = 2 \, dt \) or \( dt = \frac{du}{2} \). Substitute these into the integral: \[ \frac{1}{2} \int \frac{\frac{du}{2}}{\frac{u}{2} \sqrt{u^2-1}} \].

Step 3: Simplify the Expression

Simplify the expression from Step 2: \( \frac{1}{4} \int \frac{du}{u \sqrt{u^2-1}} \). This integral still needs further simplification to solve.

Step 4: Trigonometric Substitution

Use the trigonometric substitution \( u = \sec \theta \), thus \( du = \sec \theta \tan \theta \, d \theta \) and \( \sqrt{u^2 - 1} = \tan \theta \). Substitute these into the integral: \( \frac{1}{4} \int \frac{\sec \theta \tan \theta \, d \theta}{\sec \theta \tan \theta} = \frac{1}{4} \int d \theta \).

Step 5: Integrate with Respect to Theta

Integrate \( \frac{1}{4} \int d \theta \) to get \( \frac{1}{4} \theta + C \), where \( C \) is the constant of integration.

Step 6: Back Substitute to Original Variable

Recall \( u = \sec \theta \) and \( u = 2t \), therefore \( \sec \theta = 2t \). Hence, \( \theta = \sec^{-1}(2t) \). Thus, the solution in terms of \( t \) is \( \frac{1}{4} \sec^{-1}(2t) + C \).

Key Concepts

Trigonometric Substitution

Trigonometric substitution is an integration technique that can simplify integrals involving square roots of quadratic expressions. This method is especially useful in integrals where the expression takes a form such as \(\sqrt{a^2 - x^2}\), \(\sqrt{x^2 - a^2}\), or \(\sqrt{x^2 + a^2}\). By substituting a trigonometric identity, we can transform the integral into a more manageable form.

• For expressions like \(\sqrt{a^2 - x^2}\), the substitution is \(x = a \sin\theta\).
• For \(\sqrt{x^2 - a^2}\), use \(x = a \sec\theta\).
• For \(\sqrt{x^2 + a^2}\), consider \(x = a \tan\theta\).

In the given exercise, we apply the trigonometric substitution with \(u = \sec \theta\) since \(\sqrt{u^2 - 1} = \tan \theta\). This technique simplifies the original complex expression to an easy-to-integrate form. Once solved, we need to remember to convert back to the original variable by back-substituting our trigonometric results.

Definite and Indefinite Integrals

When dealing with integrals, it's important to recognize whether you're handling a definite or indefinite integral. The difference impacts both the approach and the solution format of calculations.

• Indefinite integrals do not have boundaries, and their result includes a constant of integration, \(C\). For example, \int f(x) \, dx = F(x) + C \.
• Definite integrals include specific upper and lower bounds, providing a numerical value representing the area under the curve between these points. They do not include a constant, for example, \int_a^b f(x) \, dx = F(b) - F(a) \.

In our exercise, the focus is on indefinite integration, as there are no specified limits attached to the integral. This requires us to include the constant \(C\) in our final solution output.

Substitution Method

The substitution method is a fundamental technique to simplify integrals where direct integration might be complex. It's akin to applying the chain rule in reverse and involves changing the variable of integration.

• Identify a part of the integral that can be substituted with a single variable, usually to simplify a more complex expression.
• Choose your substitution, say \(u = g(x)\), then express \(dx\) in terms of \(du\).
• Replace all instances of the original variable in the integral with the new substitution variable.
• Integrate with respect to the new variable \(u\).
• Finally, substitute back any expressions involving the original variable to return to the initial context.

In the exercise provided, we simplify the integral using a substitution \(u = 2t\). By choosing this substitution, we convert the expression in terms of \(u\), simplifying the variables and making the integral more straightforward to solve. This step is critical before proceeding with other techniques like trigonometric substitution.