Question

In the following exercises, compute the antiderivative using appropriate substitutions. \(\int \frac{\sec ^{-1}\left(\frac{t}{2}\right)}{|t| \sqrt{t^{2}-4}} d t\)

Short answer

The antiderivative is \( \frac{1}{2} (\sec^{-1}(\frac{t}{2}))^2 + C \).

Step-by-step solution

Identify the Substitution

To solve the integral \( \int \frac{\sec ^{-1}\left(\frac{t}{2}\right)}{|t| \sqrt{t^{2}-4}} \, dt \), we recognize \( \sec^{-1}\left(\frac{t}{2}\right) \) as an inverse trigonometric function that suggests using substitution. We will use \( t = 2 \sec(u) \), which implies \( \sec^{-1}\left( \frac{t}{2} \right) = u \). This substitution simplifies the integrand significantly.

Compute the Derivatives

Given the substitution \( t = 2 \sec(u) \), find the differential \( dt \). By differentiating, we get \( dt = 2 \sec(u) \tan(u) \, du \). This will help us rewrite the integral in terms of \( u \).

Simplify the Integral

Substitute \( t = 2 \sec(u) \) and \( dt = 2 \sec(u) \tan(u) \, du \) into the integral. The expression \( \sqrt{t^2 - 4} \) becomes \( \sqrt{(2 \sec(u))^2 - 4} = \sqrt{4 \sec^2(u) - 4} = 2 \sqrt{\sec^2(u) - 1} = 2 |\tan(u)| \). The integral now becomes \( \int \frac{u}{2 \sec(u) \cdot 2 |\tan(u)|} \cdot 2 \sec(u) \tan(u) \, du \).

Simplify Further and Integrate

Upon simplifying, the integral \( \int \frac{u \cdot 2 \sec(u) \tan(u)}{4 \sec(u) |\tan(u)|} \, du \) reduces to \( \int \frac{u \tan(u)}{|\tan(u)|} \, du = \int u \, du \), because the absolute value cancels out \( \tan(u) \) due to consistent sign within the relevant limits. Integrate \( u \, du \) to get \( \frac{u^2}{2} + C \).

Substitute Back to Original Variable

Since \( u = \sec^{-1}\left( \frac{t}{2} \right) \), substitute back to get the antiderivative in terms of \( t \). The integral becomes \( \frac{1}{2} (\sec^{-1}\left( \frac{t}{2} \right))^2 + C \).

Conclusion

The antiderivative of the original integral \( \int \frac{\sec ^{-1}\left(\frac{t}{2}\right)}{|t| \sqrt{t^{2}-4}} \, dt \) is \( \frac{1}{2} (\sec^{-1}\left( \frac{t}{2} \right))^2 + C \).

Key concepts

Substitution Method

The substitution method is a powerful tool for solving integrals, especially when dealing with complex functions. In our specific exercise, it involves changing the variable of integration to simplify the integral. This process is akin to solving a puzzle by rearranging its pieces to make it more comprehensible. To effectively use this method, the goal is to replace a portion of the integrand with a new variable that simplifies the expression.

Here are the steps you typically follow in the substitution method:

• Identify a part of the integrand that can be substituted. It often involves a function and its derivative.
• Set this part equal to a new variable, say, \( u \).
• Differentiate \( u \) with respect to \( x \) to find \( du \), then solve for \( dx \).
• Substitute \( u \) and \( du \) back into the integral, simplifying it.
• Perform the integration with respect to \( u \).
• Finally, substitute back the original variable to finish the problem.

Using substitution allows us to turn an intimidating integral into something manageable by performing these steps carefully.

Inverse Trigonometric Functions

Inverse trigonometric functions are essential when integrating expressions involving trigonometric identities. They allow us to reverse trigonometric functions, enabling us to evaluate integrals more easily. These functions include \( \sin^{-1}(x) \), \( \cos^{-1}(x) \), and \( \sec^{-1}(x) \), among others. In our exercise, \( \sec^{-1} \left( \frac{t}{2} \right) \) is the focal point.

These functions often arise when manipulating the integrands, specifically when using trigonometric identities. Here's why they are crucial in integration:

• They provide a way to solve integrals that are not straightforward by expressing variables in terms of angles.
• They help in converting complex integral calculus problems into algebraic ones.
• Using these functions, you can often find simpler antiderivatives.

For integrals with inverse trigonometric functions, understanding their derivatives is crucial, as they can simplify and guide you toward a correct substitution.

Integrals Involving Trigonometric Identities

Trigonometric identities are vital when computing integrals, especially when dealing with complex trigonometric expressions. In the case of our exercise, these identities are used to simplify both the integrand and the substitution process.

Here's how trigonometric identities assist in integration:

• They simplify expressions, making them more manageable for computation.
• They reveal connections between different trigonometric functions, which can lead to easier integration paths.
• They often reduce the number of algebraic manipulations needed, thereby decreasing the overall complexity of the integral.

Specifically, in the given problem, the identity \( \sec^2(u) - 1 = \tan^2(u) \) was used. This identity was instrumental in simplifying the term \( \sqrt{t^2 - 4} \) into \( 2 |\tan(u)| \). This simplification made integrating the function far more straightforward by reducing it to a standard form.

Understanding and being able to apply these identities effectively allows one to tackle intricate integrals with confidence and efficiency.