Question

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

Short answer

The integral simplifies by substitution of \(u=t^2\) and rewriting in terms of \(u\); further steps involve complex simplifications potentially solved using algebra systems.

Step-by-step solution

Identify the substitution

In the integral \(\int \frac{\operatorname{tsec}^{-1}(t^2)}{t^2 \sqrt{t^4-1}} dt\), we start by choosing a substitution to simplify the integral. Notice that the expression under the radical \(t^4 - 1\) suggests the substitution \(u = t^2\). This substitution simplifies \(t^4 - 1\) to \(u^2 - 1\).

Differentiate the substitution

Differentiate \(u = t^2\) to find \(du\). We have \(\frac{du}{dt} = 2t\), therefore \(du = 2t \, dt\). Solving for \(dt\), we get \(dt = \frac{du}{2t}\).

Rewrite the integral in terms of u

Substitute \(t^2 = u\) and \(dt = \frac{du}{2t}\) into the integral: \[\int \frac{\operatorname{tsec}^{-1}(u)}{u \sqrt{u^2 - 1}} \cdot \frac{du}{2t}\]. Since \(t = \sqrt{u}\), substitute \(t = \sqrt{u}\) back to simplify: \[\int \frac{\operatorname{tsec}^{-1}(u)}{u \sqrt{u^2 - 1}} \cdot \frac{du}{2 \sqrt{u}}\].

Simplify the integral

Simplify the integral further: \[\int \frac{\operatorname{tsec}^{-1}(u)}{u^{3/2} \sqrt{u^2 - 1}} \cdot \frac{1}{2} du = \frac{1}{2} \int \frac{\operatorname{tsec}^{-1}(u)}{u^{3/2} \sqrt{u^2 - 1}} du\].

Solve the integral

The integral still appears complex, and further simplifications or a specific technique like trigonometric substitution can apply. However, since precise simplification may not lead to a closed form easily, this step often requires computer algebra systems for exact evaluation or recognizing integral forms in tables, leading to solutions involving inverse trigonometric expressions and multipliers.

Key concepts

Antiderivative

An antiderivative, also known as an indefinite integral, is a fundamental concept in calculus. It operates as the reverse of differentiation. If you differentiate an antiderivative function, you will retrieve the original function back. This relationship helps in calculating areas under curves, among other applications. The notation for the antiderivative is given by the integral symbol: \(\int f(x) \, dx\), where \(f(x)\) is the function to integrate.

Understanding the nuances of finding an antiderivative is critical, as it requires recognizing various function types and selecting appropriate integration techniques. The aim is to find a function whose derivative will result back in the original function given in the integrand. This is why sometimes direct integration isn't possible, and instead, we employ methods such as substitution or integration by parts.

Integration techniques

Integration techniques are strategies employed to compute integrals, which are not straightforward. Different techniques are helpful depending on the form of the function involved. Let's dive into some commonly used methods.

• Substitution: This method simplifies an integral by changing the variable of integration. It's useful when an integral contains a composite function. By substituting a part of the function, the integral becomes easier to solve.
• Integration by Parts: Analogous to the product rule in differentiation, this technique is used when the integrand is a product of two functions.
• Partial Fraction Decomposition: Applied to integrals involving rational functions, this technique decomposes the integrand into simpler fractions that are easier to integrate.

In the exercise example, a substitution technique is employed to tackle the radical expression \(t^4 - 1\). Substitution is particularly powerful when dealing with integrals that would be cumbersome to solve directly, assisting in turning a complicated integral into a more manageable form. Often, the right technique or combination thereof is necessary to reach a solution.

Trigonometric substitution

Trigonometric substitution is a type of substitution used in integration to handle integrals involving square roots of a quadratic form. This technique is particularly useful when the presence of a radical involving squared terms appears in the integrand.

In trigonometric substitution, you replace the variable with a trigonometric function to exploit the identity of trigonometric functions to simplify the complex expressions. For instance:

• If you encounter \(\sqrt{a^2 - x^2}\), substitute \(x = a \sin \theta\).
• If you see \(\sqrt{a^2 + x^2}\), use \(x = a \tan \theta\).
• For \(\sqrt{x^2 - a^2}\), substitute \(x = a \sec \theta\).

In the provided exercise, while a trigonometric substitution was not ultimately executed, it's a vital tool when you find yourself dealing with expressions that appear too challenging to deconstruct. Mastering this tool expands your integration toolkit, allowing you to tackle a broader array of challenging mathematical problems seamlessly.