Problem 11 - Calculus Early Transcendentals

Question

Use a table of integrals to determine the following indefinite integrals. $$\int \frac{d x}{1-\cos 4 x}$$

Step-by-Step Solution

Answer

Question: Find the indefinite integral of the given function: $$\int \frac{dx}{1-\cos 4x}$$ Answer: $$\int \frac{dx}{1-\cos 4x} = -\cot(2x) + C$$

Step 1: Write down integral using a trigonometric identity

For a start, let's rewrite the given integral using the trigonometric identity mentioned in the analysis. By doing so, we obtain: $$\int \frac{dx}{1-\cos 4x} = 2 \int\frac{dx}{2\sin^2 2x}$$

Step 2: Substitute and use the integration table

Now we will substitute \(u = 2x\), and we get: $$2 \int\frac{dx}{2\sin^2 2x} = \int\frac{du}{\sin^2 u}$$ Using the integration table, we find that the integral of \(\csc^2 x\) with respect to \(x\) is equal to \(- \cot x + C\).

Step 3: Substitute back and find the indefinite integral

Now that we have found the integral with respect to \(u\), we can substitute back and find the original indefinite integral. $$\int\frac{du}{\sin^2 u} = -\cot u + C = -\cot(2x) + C$$ So, the indefinite integral is given by: $$\int \frac{dx}{1-\cos 4x} = -\cot(2x) + C$$

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations that express one trigonometric function in terms of others. They are pivotal in simplifying and solving integrals involving trigonometric functions. In this exercise, we used the identity involving sine and cosine, specifically:

• The double angle identity for sine: \[1 - \cos 4x = 2\sin^2 2x\]

This identity allows us to convert the expression \(1 - \cos 4x\) into a term involving \(\sin\), which is much simpler to integrate.
By rewriting the integral in this way, we break down a complex expression into a more manageable form. Understanding and applying these identities are essential skills for manipulating trigonometric integrals effectively. Integrals that seem complex at first sight often have simpler forms with the right identity.

Substitution Method

The substitution method is a technique used to simplify integrals. It involves changing variables to transform a difficult integral into one that is easier to solve. In our exercise, we used the substitution method as follows:

• We substituted \(u = 2x\), thus \(du = 2dx\). Solving for \(dx = \frac{du}{2}\) then allows us to substitute in the integral.

The substitution transforms the integral into a new form: \[2 \int \frac{dx}{2\sin^2 2x} = \int \frac{du}{\sin^2 u}\]
By breaking it down, the substitution method helps reduce the complexity. This step is crucial because it often turns an integral into a standard form that we can then solve using known integral tables or rules.
It’s a powerful tool for situations where direct integration is complex or impossible, converting variable expressions into more manageable equivalents.

Integration Table

An integration table is a collection of formulas used to find the antiderivatives of various functions. It empowers you to quickly solve integrals without re-deriving integral formulas. In the given problem, after substitution, the resulting integral was:

• \(\int \frac{du}{\sin^2 u}\)

By referring to an integration table, we identify this integral as the derivative of the cosecant function, specifically:\[\int \csc^2 u \, du = -\cot u + C\]
Integration tables are essential for calculus students because they save time and effort. Instead of working out every integral from scratch, you can apply these pre-established results to solve problems more efficiently. It's important to become familiar with the functions in an integration table and understand how and when to use them correctly.