Problem 37 - Calculus Single Variable

Question

Use trigonometric identities to compute the indefinite integrals. $$ \int \tan ^{2}(x) d x $$

Step-by-Step Solution

Answer

\( \int \tan^2(x) \, dx = \tan(x) - x + C \)

Step 1: Recognize a Trigonometric Identity

Recall the trigonometric identity for tangent squared: \( \tan^2(x) = \sec^2(x) - 1 \). This identity will allow us to break down the integral into a simpler form.

Step 2: Substitute the Identity into the Integral

Substitute the identity from Step 1 into the integral: \( \int \tan^2(x) \, dx = \int (\sec^2(x) - 1) \, dx \). This separates the integral into two parts: \( \int \sec^2(x) \, dx - \int 1 \, dx \).

Step 3: Integrate Separately

Integrate each part of the expression separately. The integral of \( \sec^2(x) \) is \( \tan(x) \), and the integral of \( 1 \) is \( x \). Thus, \( \int \sec^2(x) \, dx = \tan(x) + C_1 \) and \( \int 1 \, dx = x + C_2 \).

Step 4: Combine the Results

Combine the integrals with their constants of integration. The combined integral is \( \tan(x) - x + C \), where \( C \) is the constant that represents \( C_1 - C_2 \).

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in calculus, particularly when working with integrals involving trigonometric functions. These are equations that are true for all values of a variable within certain domains, establishing relations between trigonometric functions. In this exercise, the key identity used is \( \tan^2(x) = \sec^2(x) - 1 \). This identity helps us to transform the integral of \( \tan^2(x) \) into a more manageable form.

• Pythagorean Identities: These relate \( \,\sin(x), \,\cos(x) \,\) and their squares, such as \( \sin^2(x) + \cos^2(x) = 1 \).
• Reciprocal Identities: Such as \( \,\tan(x) = \frac{\sin(x)}{\cos(x)} \,\) and \( \sec(x) = \frac{1}{\cos(x)} \), these identities show how one function can be expressed in terms of others.
• Other Transformations: Using these identities can help to simplify complex trigonometric expressions, making integration feasible.

Whenever faced with a trigonometric integral, identifying and applying the right identity is the first step. It transforms complex expressions into simpler ones, opening the path for integration. The identity \( \tan^2(x) = \sec^2(x) - 1 \) in our solution allows us to break down the problem into a sum of simple integrals that are readily solvable.

Integration Techniques

Integration techniques are strategies used to integrate functions that are not easily integrable at first glance. In problems like integrating \( \tan^2(x) \), recognizing patterns and applying identities is a crucial part of the process. The selected technique in this exercise involves the substitution strategy and using basic integral formulas.

• Substitution Method: Identify expressions that can simplify. Substitute them using trigonometric identities.
• Basic Integral Forms: Recognize standard integrals such as \( \int \sec^2(x) \, dx = \tan(x) + C \) and \( \int 1 \, dx = x + C \).
• Splitting Integrals: Split a complex integral into simpler parts, making it easier to solve them separately.

In our case, by substituting \( \sec^2(x) - 1 \) for \( \tan^2(x) \), the integral becomes a sum of basic integrals. Mastery in recognizing when and how to apply these techniques can greatly ease the challenge of integration.

Tangent and Secant Functions

Tangent and secant functions are critical in both trigonometry and calculus. These functions have distinct properties and integral formulas that are useful in solving problems involving trigonometric forms.

• Tangent Function \( \tan(x) \): This function represents the ratio \( \frac{\sin(x)}{\cos(x)} \). It has a periodic nature and is important in integrals, especially with \( \sec^2(x) \).
• Secant Function \( \sec(x) \): Known as the reciprocal of cosine, \( \sec(x) = \frac{1}{\cos(x)} \), this function appears in identities and integrals due to its relationship with tangent squares \( \sec^2(x) = \tan^2(x) + 1 \).
• Integrations: The integration of \( \sec^2(x) \) directly results in \( \tan(x) \), illustrating their intertwined behaviors and importance in solving trigonometric integrals.

Understanding these functions and their properties is essential for handling calculus problems. The ability to recognize their formats and apply their integrals allows for more streamlined problem-solving, as seen in our example where integrating \( \sec^2(x) \) locks directly into an answer.