Problem 22 - Calculus: Early Transcendentals

Question

Evaluate the indefinite integral. \( \displaystyle \int \sin x \sin(\cos x) \, dx \)

Step-by-Step Solution

Answer

The integral evaluates to \( \cos(\cos x) + C \).

Step 1: Identify Integration Technique

For the given integral \( \int \sin x \sin(\cos x) \, dx \), we notice the function involves a composition of trigonometric functions. We can use a substitution method to simplify our integration process. Here, let's try using a substitution where \( u = \cos x \)

Step 2: Substitution

Let \( u = \cos x \). Then, the derivative \( du = -\sin x \, dx \), or \( -du = \sin x \, dx \). Substitute these into the integral.

Step 3: Substitute and Simplify the Integral

After substitution, the integral becomes \( \int \sin x \sin(\cos x) \, dx = \int \sin(u) (-du) \). This simplifes to \( -\int \sin(u) \, du \).

Step 4: Integrate with Respect to u

To integrate \( -\int \sin(u) \, du \), we recall the integral of \( \sin u \) is \( -\cos u \). Thus, the integral becomes \( -(-\cos u) + C = \cos u + C \).

Step 5: Substitute Back to Original Variable

Replace \( u \) with \( \cos x \) to return to the original variable. Thus, the final solution is \( \int \sin x \sin(\cos x) \, dx = \cos(\cos x) + C \), where \( C \) is the constant of integration.

Key Concepts

Substitution Method

The substitution method is a powerful technique for evaluating integrals, especially when dealing with compositions of functions. It's particularly useful when you can identify a part of the integrand as a derivative of another function. In essence, you "substitute" a portion of the integrand with a single variable to make the integration process smoother.

In our example, we noted that the integral \( \int \sin x \sin(\cos x) \, dx \) contains a nested function \( \cos x \). By setting \( u = \cos x \), we can rewrite the integral in terms of \( u \) instead of \( x \). Calculate the derivative of \( \cos x \) to express \( dx \) in terms of \( du \): \( du = -\sin x \, dx \). Then, the substitution transforms the integral into a simpler integral in terms of \( u \): \( -\int \sin(u) \, du \).

This method not only simplifies the integral but also makes it more approachable by converting it into a standard form that can be integrated directly. It is a crucial step in handling complex integrals, where traditional methods might not be feasible.

Trigonometric Integration

When it comes to integrating trigonometric functions, recognizing and using specific identities can significantly simplify the process. Trigonometric integration often involves functions like sine, cosine, and tangent, among others. These functions can be trickier to integrate due to their oscillatory nature.

The integral we worked on, \( \int \sin x \sin(\cos x) \, dx \), involves two sine functions, one of which is nested inside the other. This compound structure calls for careful treatment, making substitution an ideal method for this case. By recognizing that one of the sine terms is actually a function of the cosine, we can handle it more effectively with substitution, as seen in our example.

However, it's essential to be familiar with basic trigonometric identities such as \( \sin^2 x + \cos^2 x = 1 \), or integration rules like \( \int \sin x \, dx = -\cos x + C \) and \( \int \cos x \, dx = \sin x + C \). These identities help in rewriting or simplifying integrals that at first might seem difficult to solve.

Integration Techniques

Integration is a fundamental concept in calculus, directly used for finding areas, volumes, central points, and many useful things. The process of integration can often be simplified with specific techniques. Here are some specific techniques that can aid in solving complex integrals:

• Substitution: As we've already discussed, substitution simplifies the integrand by changing variables, making it an effective approach when dealing with composite functions.
• Integration by Parts: Useful when the integrand is a product of two functions. It is based on the product rule for differentiation and can transform a challenging integral into a more manageable form.
• Partial Fraction Decomposition: This is great for rational functions, breaking them down into simpler fractions that are easier to integrate.
• Trigonometric Substitutions: Especially useful for integrals involving \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), or \( \sqrt{x^2 - a^2} \) where trigonometric identities can simplify the expressions.

In the integral \( \int \sin x \sin(\cos x) \, dx \), combining substitution with trigonometric integration is key. Mastery of these techniques is essential for solving a wide variety of integrals efficiently. Don't hesitate to mix and match these strategies based on the problem at hand.