Problem 62 - Calculus Early Transcendentals

Question

Use a change of variables to evaluate the following integrals. $$\int \sec 4 w \tan 4 w d w$$

Step-by-Step Solution

Answer

Question: Evaluate the integral of the product of the secant and tangent functions of 4w with respect to w. Answer: \(\frac{1}{4}(\sec(4w) + C)\)

Step 1: Select substitution

Let's choose the substitution: $$u = 4w$$ Then, find the differential for our new variable, u: $$du = 4dw$$

Step 2: Substitute

Now, we will make the substitutions in the integral so we can work with a simpler expression: $$\int \sec 4w \tan 4w dw = \frac{1}{4}\int \sec u \tan u du$$ (Notice how we divided by 4 to account for the \(du = 4dw\) relationship.)

Step 3: Apply integral identity

We recognize that this integral fits the identity: $$\int \sec u \tan u du = \sec u + C$$ where C is the constant of integration.

Step 4: Evaluate the integral

Applying the identity to our integral: $$\frac{1}{4}\int \sec u \tan u du = \frac{1}{4}(\sec u + C)$$

Step 5: Reverse substitution

Now, we need to reverse our substitution in order to express the result in terms of the original variable, w. Recall that our substitution was: $$u = 4w$$ So, our secant term becomes: $$\sec u = \sec(4w)$$ Thus, our final result is: $$\frac{1}{4}(\sec(4w) + C)$$

Key Concepts

Change of Variables

The technique of changing variables, often known as "u-substitution," is a powerful method in calculus for simplifying complex integrals. The central idea is to transform an integral into a more manageable form by substituting a variable. This method is particularly useful when you encounter an integral that arises from the chain rule in differentiation.

Here's how it works: You choose a new variable, typically denoted as 'u', to replace a more complicated part of the integrand. In our exercise:

• We set $u = 4w$.
• This means wherever '4w' appears in the integral, it's replaced by 'u'.

The key step is finding the differential of the new variable, which helps us replace the original differential. In this case:

• The differential $du=4dw$ replaces $dw$ leading to mapping between differential space.

This transformation allows you to convert the given integral into a simpler form, which is often easier to evaluate or recognize. In problems like ours, identifying the appropriate substitution is crucial and often requires practice and experience to spot immediately.

Trigonometric Integrals

Trigonometric integrals involve the integration of functions containing trigonometric functions, such as sine, cosine, tangent, and secant. These functions often appear in oscillating behaviors like waves, circles, and pendulums. In these integrals, you frequently use identities or substitutions to simplify the integration process.

In the given exercise, we encountered functions \(\sec(4w)\) and \(\tan(4w)\). The integration of trigonometric functions involving secant and tangent is especially convenient when we use standard formulas. For instance:

• The integral \(\int \sec(u) \tan(u) \ du = \sec(u) + C\) is a standard identity.

Such identities reduce the work needed to solve complex integrals and help apply the correct integration technique swiftly. Recognizing these standard forms is essential in solving trigonometric integrals effectively.

By understanding these identities, you can transform challenging expressions into ones that are straightforward to integrate because they fit well-known patterns.

Substitution Method

The substitution method is another term for the change of variables and is widely used due to its simplicity and effectiveness. It's particularly helpful for evaluating integrals, either indefinite or definite, which are difficult to handle in their presented form. This technique can be broken down into systematic steps:

• Identify a section of the integrand that can be replaced with a single variable, `u`.
• Compute the differential of `u` (noted as `du`) and determine how it relates to the current variable.
• Perform the substitution to transform the integral into terms of 'u'.
• Integrate the resulting expression in terms of 'u'.
• Finally, reverse the substitution, expressing the integral in terms of the original variable again.

In our example, substituting $4w$ with $u$ simplifies the integral involving trigonometric functions. After performing the integration with respect to $u$, reversing the substitution ensures that the final answer is in terms of the original variable, $w$.

This method streamlines integration by reducing complex compositions into simpler algebraic forms, reflecting the reverse of differentiation's chain rule.