Problem 14 - Calculus Early Transcendentals

Question

Evaluate the following integrals. Include absolute values only when needed. \(\int \tan 10 x d x\)

Step-by-Step Solution

Answer

Question: Evaluate the integral \(\int \tan(10x) dx\). Answer: \(\int \tan(10x) dx = -\dfrac{1}{10}\ln|\cos(10x)| + C\)

Step 1: Recognize the given integral

We are given the integral \(\int \tan(10x) dx\). It's the integration of the tangent function with a coefficient inside the argument.

Step 2: Rewrite the tangent function

Using the trigonometric identity, we can rewrite the tangent function as follows: \(\tan(10x) = \dfrac{\sin(10x)}{\cos(10x)}\) The integral becomes: \(\int \dfrac{\sin(10x)}{\cos(10x)} dx\)

Step 3: Use substitution method

Let's use the substitution method to integrate. Choose the substitution \(u = \cos(10x)\). Then, differentiate \(u\) with respect to \(x\): \(du = -10\sin(10x) dx\) Rearranging for \(dx\), we have: \(dx = \dfrac{1}{-10\sin(10x)} du\) Now, substitute \(u\) and \(dx\) in the integral: \(\int \dfrac{\sin(10x)}{u} \cdot \dfrac{1}{-10\sin(10x)} du = \int \dfrac{1}{-10u} du\)

Step 4: Integrate with respect to u

Now, integrate with respect to \(u\): \(\int \dfrac{1}{-10u} du = -\dfrac{1}{10}\int \dfrac{1}{u} du\) Using the properties of the natural logarithm, we have: \(-\dfrac{1}{10}\int \dfrac{1}{u} du = -\dfrac{1}{10}\ln|u| + C\)

Step 5: Substitute back

Now, substitute back for \(u = \cos(10x)\): \(-\dfrac{1}{10}\ln|\cos(10x)| + C\) Thus, the integral of the given function is: \(\int \tan(10x) dx = -\dfrac{1}{10}\ln|\cos(10x)| + C\)

Key Concepts

Indefinite Integral

At the heart of calculus lies the concept of the indefinite integral. It represents the antiderivative of a function and is used to find a function that describes the area under the curve for a given mathematical expression. When we write the indefinite integral of a function, it takes the general form \( \int f(x) dx \), where \( f(x)\) is the function to be integrated and \( dx \) indicates integration with respect to variable \( x \). Because the integral defines an antiderivative, it always includes a constant of integration, \( C \) — hence why the solution to an indefinite integral is expressed as a family of functions rather than a single function.

When tackling integrals, especially indefinite ones, we often deal with unknown bounds. This characteristic distinguishes them from definite integrals, which calculate a specific value over a defined interval. Indefinite integrals are pivotal in various areas such as physics, engineering, and economics, where they help calculate quantities like displacement, potential energy, and accumulated growth.

Trigonometric Integration

Trigonometric integration involves finding the indefinite integral of functions involving trigonometric functions such as sine, cosine, tangent, among others. These functions are periodic and wave-like, commonly appearing in problems related to oscillations, waves, and rotation movements.

Integrating trigonometric functions often requires the use of trigonometric identities to simplify the expression before integration. One such identity is that \(\tan x = \frac{\sin x}{\cos x}\). In the case of our exercise, the integral of \(\tan(10x)\) is approached by rewriting it using this identity, which allows the problem to be framed in a more manageable form for integration techniques like substitution. The familiarity with various trigonometric identities greatly facilitates the process of integrating these types of functions.

Applying these identities purposefully can transform a complex integral into a simple one. As seen with the exercise, without this step of simplification, you might find yourself stuck or taking much longer to solve the problem.

Integration Techniques

There is a suite of techniques in the calculus toolbox to solve integrals, each with its own scenarios where it's most effectively applied. The basic techniques include the power rule, integration by parts, and trigonometric integration, while more advanced techniques involve partial fractions, trigonometric substitution, and integration by series. When presented with a complex integral, the goal is to transform it into a simpler form that can be integrated directly.

For the integral presented in our exercise, we are primarily concerned with the substitution method, also called 'u-substitution'. This technique is akin to reverse chain rule—it simplifies the integral by transforming it to a new variable, \( u \), where the integration becomes more straightforward. Finding the right substitution is key; in some cases, like trigonometric integrations, a good substitution can be guided by existing trigonometric identities.

U-Substitution

When it comes to the 'u-substitution' method, the crux of the technique lies in choosing a part of the original function, which when substituted with \( u \), will make the integral more manageable. The choice of \( u \), therefore, is not arbitrary, but strategic. After selecting \( u \), we differentiate it with respect to \( x \) to find \(\frac{du}{dx}\) or conversely \( dx \) in terms of \( du \).

In the given exercise's solution, \( u = \cos(10x) \) is chosen as the substitution, and through differentiation, we derive \( dx \) in terms of \( du \) which is then used to replace \( dx \) in the original integral. After integration and simplification, it's crucial to 'substitute back', replacing \( u \) with the original function, resulting in a solution in terms of the original variable, \( x \). U-substitution is an elegant method that simplifies many intimidating integrals into forms that are significantly more tractable.