Problem 348 - Calculus Volume 1

Question

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. $$\int \tan (2 x) d x$$

Step-by-Step Solution

Answer

The integral is \(-\frac{1}{2} \ln|\cos(2x)| + C\).

Step 1: Rewrite Tan Function

Using the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \), we rewrite \( \tan(2x) \) as \( \frac{\sin(2x)}{\cos(2x)} \).

Step 2: Substitution

Let \( u = \cos(2x) \), then \( du = -2\sin(2x) dx \), hence \( \sin(2x) dx = -\frac{1}{2} du \).

Step 3: Change Integral in Terms of u

Substitute the expressions found in Step 2 back into the integral: \( \int \tan (2x) dx = \int \frac{\sin(2x)}{\cos(2x)} dx = \int \frac{1}{u} \left( -\frac{1}{2} du \right) = -\frac{1}{2} \int \frac{1}{u} du \).

Step 4: Integrate

The integral of \( \frac{1}{u} \) is \( \ln|u| + C \). Thus, \( -\frac{1}{2} \int \frac{1}{u} du = -\frac{1}{2} \ln|u| + C \).

Step 5: Back Substitution

Replace \( u \) with \( \cos(2x) \) to get the final result: \( -\frac{1}{2} \ln|\cos(2x)| + C \).

Key Concepts

Substitution Method

The substitution method is a technique used in calculus to simplify an integral by changing its variable. This often makes the integral easier to solve. To apply the substitution method, you choose a part of the integrand (the function to be integrated) and set it equal to a new variable, usually denoted as \(u\). In the example \( \int \tan(2x) \, dx \), we choose \(u = \cos(2x)\). This substitution is effective because the derivative of \(\cos(2x)\) is related to \(\sin(2x)\), which appears in our integral. Key Steps:

• Identify a substitution that simplifies the integrand.
• Compute the differential of the substitution, i.e., \( du \).
• Rewrite the entire integral in terms of \(u\) and \(du\).
• Perform the integration on the simpler expression.
• Substitute back the original variable to obtain the solution.
  For this integral, substituting \(u = \cos(2x)\) led to an easier integration since \(\int \frac{1}{u} \, du = \ln|u| + C\).

This method turns a potentially complex problem into a manageable one.

Logarithmic Integration

Logarithmic integration is a technique used to evaluate integrals of the form \(\int \frac{1}{u} \, du\), which result in a logarithmic function of \(u\). Within our example, the substitution \(u = \cos(2x)\) results in the integral \(-\frac{1}{2}\int \frac{1}{u} \, du\). The integral of \(\frac{1}{u}\) is the natural logarithm, giving us the result \(-\frac{1}{2}\ln|u| + C\).Important Points:

• The logarithmic form \(\int \frac{1}{u} \, du\) is important because many real-world problems can be formulated in this way.
• Always include the absolute value in the logarithmic integration, \(\ln|u|\), to ensure correctness over all possible values of \(u\).
• Don’t forget the constant of integration \(C\), which accounts for any vertical shift in the indefinite integration.
  In our example, once you complete substitution and simpler integration, transform back to the original variable results in the expression \(-\frac{1}{2}\ln|\cos(2x)| + C\).

This shows how complex trigonometric integrals can lead to simpler logarithmic results.

Trigonometric Identities

Trigonometric identities play a crucial role in solving integrals involving trigonometric functions. They allow us to rewrite expressions in more manageable forms. In our exercise, we use the identity \(\tan(x) = \frac{\sin(x)}{\cos(x)}\) to transform \(\tan(2x)\) in terms of sine and cosine, making it suitable for substitution.Helpful Trigonometric Identities:

• Basic Identity: \(\tan(x) = \frac{\sin(x)}{\cos(x)}\).
• Double Angle Formula: Can be useful for more complex trigonometric integrals.
• Reciprocal Identity: For transforming secant, cosecant, or other related functions.
  In practice, identify the identity that aligns with the integral’s structure to simplify it.

Using these identities helps to bridge the complex forms of trigonometric expressions. By applying the correct identity, we make integration feasible and more straightforward, as was demonstrated in our solution with \(\tan(2x)\).