Question

In the following exercises, use appropriate substitutions to express the trigonometric integrals in terms of compositions with logarithms. $$ \int \frac{\sin (3 x)-\cos (3 x)}{\sin (3 x)+\cos (3 x)} d x $$

Short answer

\( \frac{1}{2}\tan^2(3x/2) - \tan(3x/2) + C. \)

Step-by-step solution

Identify a suitable substitution

Notice that the integral has the form of a trigonometric function fraction that can be simplified using a substitution. Let's use the substitution \( u = \tan(3x/2) \). Thus, the identity \( \sin(3x) + \cos(3x) = \sqrt{2} \cos(3x/2 - \pi/4) \) can be used, rooted in sum-to-product identities.

Express trigonometric functions in terms of substitution

Using the substitution \( u = \tan(3x/2) \), recall that \( \sin(3x) = 2\sin(3x/2)\cos(3x/2) \) and \( \cos(3x) = \cos^2(3x/2) - \sin^2(3x/2) \). Write both \( \sin(3x) \) and \( \cos(3x) \) in terms of \( u \).

Differentiate the substitution to find \( dx \)

Differentiate \( u = \tan(3x/2) \) to find \( du/dx = \frac{3}{2}\sec^2(3x/2) \). Thus, \( dx = \frac{2}{3}\cos^2(3x/2)\ du \).

Rewrite the integral in terms of \( u \)

Substitute \( u = \tan(3x/2) \) into the integral to obtain:\[\int \frac{\sin(3x) - \cos(3x)}{\sin(3x) + \cos(3x)} dx = \int \frac{2u/(1+u^2) - (1-u^2)/(1+u^2)}{2/(1+u^2)} \cdot \frac{2}{3}\cos^2(3x/2)\ du \]This simplifies to \( \int \left( u - 1 \right) du \), by canceling and simplifying terms.

Evaluate the integral

Evaluate the integral obtained from step 4:\[\int (u - 1) du = \int u \, du - \int du = \frac{u^2}{2} - u + C\]

Substitute back in terms of \( x \)

Recall \( u = \tan(3x/2) \) from step 1. Substitute back into the evaluated integral:\[\frac{u^2}{2} - u + C = \frac{1}{2}\tan^2(3x/2) - \tan(3x/2) + C.\]

Key concepts

Trigonometric Substitution

Trigonometric substitution is a useful technique when dealing with integrals involving trigonometric functions. It simplifies complex expressions and turns them into more manageable forms.
In this exercise, we start by recognizing that the integral contains a fraction of trigonometric functions. To solve it, we need a clever substitution. By using the potential substitution, such as \( u = \tan\left(\frac{3x}{2}\right) \), we can simplify the integral substantially. This is because tangent functions relate conveniently to sines and cosines, which are parts of our fraction.
Furthermore, this substitution allows the conversion of trigonometric functions into algebraic expressions, making the integration process easier and more straightforward.

• Choose the substitution based on the trigonometric function involved.
• Convert all trigonometric expressions in the integral to the substituted variables.
• Derive \( dx \) in terms of the substitution variable to enable integration.

Integration Techniques

Integration techniques are central to calculus, especially when dealing with complex integrals that involve trigonometric functions.
When confronted with our integral, \( \int \frac{\sin(3x) - \cos(3x)}{\sin(3x) + \cos(3x)} \), straightforward methods of integration don't suffice. This calls for additional techniques, such as substitution combined with simplifying identities.
The goal is to transform the integral into a form that we can easily integrate, often by reducing it to a polynomial form or simplifying the denominator or numerator.
By substituting and simplifying, we can then evaluate the integral and eventually revert our variable substitution back to the original terms.

Sum-to-Product Identities

Sum-to-product identities are a set of trigonometric identities that help simplify sums and differences of sine and cosine into products. These identities are highly beneficial when simplifying integrals that involve sums or differences of trigonometric functions.
In our exercise, we encountered the expression \( \sin(3x) + \cos(3x) \), which can be rewritten using these identities. For example, we use:

• \( \sin(A) + \cos(A) = \sqrt{2} \cos(A - \pi/4) \)

This can transform complicated sums into cleaner expressions involving products, making subsequent steps in solving the integral more straightforward.
This means any complex trigonometric integral can be effectively transformed by breaking down the sum of functions into simpler, integrated components.

Trigonometric Identities

Trigonometric identities are the tools that make the integration of trigonometric functions manageable. They allow us to rewrite expressions, making complex integrals more approachable.
In this integral problem, we applied several known identities. For instance:

• \( \sin(2A) = 2 \sin(A) \cos(A) \)
• \( \cos(2A) = \cos^2(A) - \sin^2(A) \)

These identities offer ways to express trigonometric functions in alternative forms, which is crucial when simplifying integrals.
Moreover, knowing how to use these identities helps in identifying potential substitutions necessary to ease integration, as well as verifying the simplification process once you've solved the integral.