Problem 18 - Calculus: Early Transcendentals

Question

Find the general indefinite integral. \( \displaystyle \int \frac{\sin 2x}{\sin x}\, dx \)

Step-by-Step Solution

Answer

The integral is \( 2 \sin x + C \).

Step 1: Simplify the Integral

The given integral is \( \int \frac{\sin 2x}{\sin x} \, dx \). We start by using the double angle identity for sine, which states that \( \sin 2x = 2 \sin x \cos x \). Substitute this into the integral to get:\[ \int \frac{2 \sin x \cos x}{\sin x} \, dx \]This simplifies to \( \int 2 \cos x \, dx \) after canceling \( \sin x \) from the numerator and the denominator.

Step 2: Integrate the Simple Function

Now our integral is reduced to \( \int 2 \cos x \, dx \). Integrate \( 2 \cos x \) directly:\[ \int 2 \cos x \, dx = 2 \int \cos x \, dx \]The integral of \( \cos x \) is \( \sin x \), so:\[ 2 \int \cos x \, dx = 2 \sin x + C \]Thus, the integral evaluates to \( 2 \sin x + C \).

Key Concepts

Trigonometric Integrals

Trigonometric integrals involve finding the integral of functions composed of trigonometric functions like sine, cosine, tangent, and their combinations. In our specific exercise, we dealt with an integral involving sine functions:

• The original integral is \( \int \frac{\sin 2x}{\sin x}\, dx \).
• Understanding and identifying the structure of such functions is key to simplifying them and applying appropriate identities.
• Trigonometric identities often help reduce these complex expressions into simpler forms that are easier to integrate.

By simplifying the function through trigonometric identities, we arrived at a much simpler integral, \( \int 2 \cos x\, dx \), which can be solved directly.

Integration Techniques

Solving integrals efficiently often involves recognizing patterns or employing specific techniques or transformations to simplify the expression. For the problem at hand:

• We applied a substitution based on the known trigonometric identity \( \sin 2x = 2 \sin x \cos x \). This transformation converts the integral into a more manageable form.
• Once simplified, the integral of \( 2 \cos x \) is much easier to deal with, allowing us to integrate directly.

In integration, recognizing when to use these techniques is crucial because they transform a complex problem into a simple one. The use of trigonometric identities is a powerful tool in this regard, helping simplify the integrand by exploiting known mathematical truths.

Trigonometric Identities

Trigonometric identities are mathematical equations that relate different trigonometric functions. They are the tools we use to simplify integrals involving trigonometric functions:

• In our exercise, the key identity used was \( \sin 2x = 2 \sin x \cos x \).
• By substituting this identity into the original integral, we simplified the problem considerably.
• This step is crucial, as it often converts an unclear function into one that can be tackled with standard integration techniques.
• Other identities, such as \( \cos^2 x + \sin^2 x = 1 \) or Pythagorean identities, might also be useful for different types of trigonometric integrals.

A solid understanding of these identities is essential when tackling trigonometric integrals, as they allow for substitutions and simplifications that make integration possible.