Problem 81 - Thomas Calculus

Question

Evaluate each integral in Exercises \(71-82\) by using any technique you think is appropriate. $$ \int \sec ^{2} t \tan (\tan t) d t $$

Step-by-Step Solution

Answer

\( \ln| \sec(\tan t) | + C \)

Step 1: Identify the inner function

First, let's identify the inner function for potential substitution. Given the integral \( \int \sec^2 t \tan(\tan t) \, dt \), the inner function within the "tan" inside \( \tan(\tan t) \) can be \( u = \tan t \).

Step 2: Compute the derivative of the inner function

The next step is to compute the derivative of \( u \) with respect to \( t \). Since \( u = \tan t \), we have \( \frac{du}{dt} = \sec^2 t \). This matches part of the integral and is a clue that substitution is a good technique here.

Step 3: Substitute the inner function into the integral

Replace \( \tan t \) with \( u \) in the given integral. The differential \( dt \) becomes \( \frac{du}{\sec^2 t} = du \). Thus, the integral becomes: \[ \int \sec^2 t \cdot \tan(u) \cdot \frac{du}{\sec^2 t} = \int \tan(u) \, du \]

Step 4: Integrate the substituted integral

Now integrate \( \int \tan(u) \, du \). The standard integral of \( \tan(u) \) is \( \ln|\sec(u)| + C \). This uses the identity \( \int \tan x \, dx = \ln|\sec x| + C \).

Step 5: Replace the inner variable with the original variable

Since we used the substitution \( u = \tan t \), replace \( u \) with \( \tan t \) in the integral result. The final result is: \[ \ln| \sec(\tan t) | + C \]

Step 6: State the final result

After replacing back to the original variable, the integral of \( \int \sec^2 t \tan(\tan t) \, dt \) is: \[ \ln| \sec(\tan t) | + C \]

Key Concepts

Substitution Method

The substitution method, also known as u-substitution, is a powerful integration technique used to simplify complex integrals. It's particularly useful when dealing with composite functions. The basic idea behind substitution is to transform the integral into an easier form by changing variables.To apply substitution, follow these steps:

• Identify the inner function. In our example, this is found within the nested function, giving us the substitution variable. Here, it was chosen as \( u = \tan t \).
• Compute the derivative of the substitution variable \( u \) with respect to the original variable \( t \). This provides \( du = \sec^2 t \, dt \).
• Replace the original variables in the integral with the new variables, transforming the integral into a more familiar form.
• Perform the integration using the new variable.
• Return to the original variable by back-substituting the initial substitution expression.

The substitution method simplifies integration significantly when chosen correctly. It is essential to identify functions whose derivatives are present elsewhere in the integrand.

Trigonometric Integration

Trigonometric integration is a technique specifically designed for integrals that include trigonometric functions. These types of integrals often require special consideration because trigonometric identities and relationships can simplify them.Trigonometric identities that are commonly used include:

• \( \sin^2 x + \cos^2 x = 1 \)
• \( 1 + \tan^2 x = \sec^2 x \)
• \( \sin 2x = 2 \sin x \cos x \)

Using these identities helps simplify integrals containing powers or products of sine, cosine, tangent, or other trig functions. In our example, recognizing that \( \sec^2 t \) is the derivative of \( \tan t \) allowed for a simpler substitution. Understanding relationships between trig functions simplifies calculations by either direct integration or after substitution.

Integration by Parts

Integration by parts is a method used when the integrand is a product of two functions. It comes from the integration formula derived from the product rule for differentiation. The integration by parts formula is:\[ \int u \, dv = uv - \int v \, du \]You choose one part of the integral to differentiate (\( u \)) and the other to integrate (\( dv \)). It is especially useful when other methods, like substitution, are not applicable.Considerations for choosing \( u \) and \( dv \):

• \( u \) should be a function that simplifies upon differentiation.
• \( dv \) should be easily integrable.

Though not directly used in our example, understanding integration by parts is valuable for handling more complex expressions where straightforward substitution isn't possible.

Integration Constants

When performing indefinite integrals, an important step is the inclusion of the integration constant, often denoted as \( C \). The integration constant arises because integration is the reverse process of differentiation, and the original function could have had any constant added to it.Why is the constant important?

• It represents an entire family of functions that differ by a constant.
• Solutions to integrals without a boundary can yield infinitely many results, which are captured by \( C \).
• In physical problems, it can represent initial conditions or other specific properties inherent to the problem's context.

In our example, after integrating, we include \( C \) in the final answer, \( \ln| \sec(\tan t) | + C \), to acknowledge this potential variance. Always remember, while the calculation process might abstract it out, including the constant ensures the solution remains complete.