Problem 26 - Calculus: One and Several Variables

Question

Calculate. (If you run out of ideas, use the examples as models.) $$\int \tan ^{3} x \sec ^{3} x d x$$.

Step-by-Step Solution

Answer

The integral \(\int \tan^3x \sec^3x dx\) can be solved by using the trigonometric identity \(1 + \tan^2x = \sec^2x\), performing the substitution \(u = \tan x\), and then integrating the simplified expression. The final result is \(\frac{1}{4}\tan^4x + \frac{1}{6}\tan^6x + C\).

Step 1: Use the trigonometric identity

Rewrite \(\sec^3x\) in terms of \(\tan^2x\), using the identity \(1 + \tan^2x = \sec^2x\). We can write \(\sec^3x\) as \(\sec^2x\cdot\sec x\) and then rewrite \(\sec^2x\) as \(1 + \tan^2x\): \[ \int \tan^3x \sec^3x dx = \int (\tan^3x)(1 + \tan^2x)\sec x dx \]

Step 2: Perform substitution

Now, we'll use the substitution \(u = \tan x\). This means that \(du = \sec^2x dx\). We can rewrite the integral in terms of u: \[ \int (u^3)(1 + u^2) du \]

Step 3: Integrate the simplified expression

Next, we will expand the expression inside the integral and then integrate each term separately: \[ \int (u^3 + u^5) du = \int u^3 du + \int u^5 du \] Now, we integrate each term: \[ \frac{u^4}{4} + \frac{u^6}{6} + C \]

Step 4: Reverse the substitution

Finally, we revert back to the original variable x, using the substitution \(u = \tan x\): \[ \frac{(\tan x)^4}{4} + \frac{(\tan x)^6}{6} + C \] This is the final result: \[ \boxed{\frac{1}{4}\tan^4x + \frac{1}{6}\tan^6x + C} \]

Key Concepts

Trigonometric Integration

When approaching integrals that involve trigonometric functions, recognizing identities and patterns can be a key strategy. In trigonometric integration, the main focus is on integrating powers of sine, cosine, tangent, and secant functions. Sometimes, these functions appear in combinations that can be simplified using trigonometric identities. For instance, the Pythagorean identity relating tangent and secant, \( 1 + \tan^2x = \sec^2x \), can transform an integral into a more manageable form.

Using these identities to manipulate the integral's expression allows for easier integration, often leading to a standard antiderivative that can be calculated directly. The choice of the identity to use depends on the given functions and their powers within the integral.

Integration by Substitution

Integration by substitution, also known as 'u-substitution', is a technique comparable to applying the chain rule in reverse. It is especially useful when an integral involves a composite function or when integrating a product of functions where one is the derivative of the other. The basic idea is to substitute a part of the expression with a new variable, simplifying the integral.

In the case of the integral of \( \tan^3x \sec^3x \), substituting \( u = \tan x \) makes the integral more approachable as the derivative of \( \tan x \) is \( \sec^2x \), which appears in the original integral. After substitution, the integral becomes a polynomial in terms of \( u \), which is straightforward to integrate. This step-by-step simplification of a complex-looking integral is the essence of the substitution technique.

Integral of Tan and Sec Functions

The integral of \( \tan \) and \( \sec \) functions can sometimes puzzle students due to the various powers these functions can be raised to. In the given integral \( \int \tan^3x \sec^3x dx \), the interaction between \( \tan \) and \( \sec \) is such that they can be expressed in terms of each other using trigonometric identities.

For integrals containing higher powers of \( \tan \) or \( \sec \) functions, a common approach is to separate out a single \( \sec x \), which pairs with \( \sec^2x \) that often results from the derivative of \( \tan x \) in the substitution step. This technique ensures that all terms within the integral are in the same variable after substitution, facilitating their integration. Understanding how \( \tan \) and \( \sec \) functions are interrelated is crucial for solving such integrals.

Antiderivatives and Indefinite Integrals

Antiderivatives and indefinite integrals represent the reverse process of differentiation. Finding an antiderivative means determining a function whose derivative equates to the given function within the integral. Indefinite integrals, symbolized by \( \int \), are essentially the general family of antiderivatives that include an arbitrary constant, \( C \), as there could be an infinite number of possible functions that satisfy the condition of being an antiderivative.

When computing the indefinite integral, each term's antiderivative is calculated and summed up, with the constant \( C \) appended at the end. For polynomial expressions after a substitution, like \( u^3 \) or \( u^5 \), their antiderivatives can be determined by increasing the power by one and dividing by the new power. The final step often involves reversing the substitution, substituting back the original variable to present the answer in its initial context.