Question

Indefinite integrals Use a change of variables or Table 5.6 to evaluate the following indefinite integrals. Check your work by differentiating. $$\int \sec 4 w \tan 4 w \, d w$$

Short answer

Question: Evaluate the indefinite integral $$\int \sec 4 w \tan 4 w \, d w$$ using a change of variables or Table 5.6. Answer: $$\int \sec 4w \tan 4w \, dw = \frac{1}{4} \sec 4w + C$$

Step-by-step solution

Identify the substitution

Let's represent the integral using a new variable, $$u = 4w$$. Next, we differentiate $$u$$ with respect to $$w$$, so we can find $$\frac{du}{dw}$$. $$\frac{du}{dw} = 4$$ Now we need to find $$dw$$ in terms of $$du$$, $$dw = \frac{1}{4}du$$ We can rewrite the integral in terms of $$u$$ as: $$\int \sec u \tan u \, \frac{1}{4}du$$ This simplifies to: $$\frac{1}{4} \int \sec u \tan u \, du$$

Integrate with respect to u

Now we can integrate $$\sec u \tan u$$ with respect to $$u$$: $$\frac{1}{4} \int \sec u \tan u \, du = \frac{1}{4} \sec u + C$$ We got this result because the integral of $$\sec u \tan u$$ with respect to $$u$$ is $$\sec u$$.

Substitute back for w

Now we substitute back $$w$$ in place of $$u$$: $$\frac{1}{4} \sec 4w + C$$ So, the indefinite integral is: $$\int \sec 4w \tan 4w \, dw = \frac{1}{4} \sec 4w + C$$

Checking the result by differentiating

We need to check our result by differentiating. The derivative of $$\frac{1}{4} \sec 4w + C$$ with respect to $$w$$ is: $$\frac{d}{dw}(\frac{1}{4} \sec 4w + C) = \sec 4w \tan 4w$$ Since the derivative matches the integrand, our result is correct.

Key concepts

Change of Variables

In calculus, the "change of variables" technique serves as a powerful tool for simplifying complex integrals. When faced with an integral that looks challenging, the change of variable approach involves transforming the variable of integration. This is done by substituting the existing variable with a new variable, often referred to as "u" substitution. The overall goal is to make the integral easier to evaluate.

Here's a quick breakdown of how it works:

• Choose a new variable: For example, if the integral involves a trigonometric function like \(4w\), we might let \(u = 4w\).
• Differentiate: By differentiating \(u = 4w\), you obtain \(\frac{du}{dw} = 4\). This helps us express \(dw\) in terms of \(du\).
• Substitute: Rewrite the integral in terms of \(u\). This involves replacing \(dw\) and adjusting the limits of integration if necessary.

Through this process, the integral often simplifies significantly, paving the way to evaluate it more easily.

Integration by Substitution

Integration by substitution is essentially a manifestation of the change of variables method and is a cornerstone technique for solving indefinite integrals. The main idea is to transform the integral into a simpler form that is easier to integrate. This is particularly useful when a function and its derivative are both present in the integral.

Here is a step-by-step way to apply it:

• Select a function for substitution: Pick a part of the integrand that could be substituted to reduce complexity, such as \(u = 4w\).
• Substitute and simplify: Transform the integral to a function of \(u\) and introduce the du term, ensuring all parts of the integral are in terms of \(u\).
• Integrate with respect to the new variable: Calculate the integral in terms of \(u\).
• Back-substitute: Replace \(u\) back in terms of the original variable to return to the original variable context.

The integration by substitution process simplifies integration by focusing on easier-to-integrate pieces. It's akin to reverse chain rule used in differentiation.

Differentiation

Differentiation is the mathematical process of finding a derivative, which essentially represents the rate at which a quantity changes. In the context of integrals and their solutions, differentiation is used as a verification step to ensure the accuracy of the integration process.

Here's how to apply differentiation to verify an integral:

• Compute the derivative: Take the derivative of the result obtained from the integration step. This involves using the differentiation rules you know, like the derivative of \(\sec(4w)\) with respect to \(w\).
• Match it with the original integrand: The derivative should match the original function you integrated. For instance, in \(\int \sec 4w \tan 4w \, dw\), taking the derivative of \(\frac{1}{4}\sec 4w + C\) should give you back \(\sec 4w \tan 4w\).
• Confirm correctness: If the derivative matches, your integration results are correct. If not, revisit the substitution or integration steps.

Using differentiation in this way acts as a double-check mechanism, confirming that the anti-derivative was computed correctly. This is crucial in ensuring that the indefinite integral solution is valid.