Problem 15 - Calculus: Early Transcendentals

Question

Evaluate the indefinite integral. \( \displaystyle \int \cos^3 \theta \sin \theta \, d\theta \)

Step-by-Step Solution

Answer

\( -\frac{\cos^4 \theta}{4} + C \)

Step 1: Identify the substitution

Since we have a composite function involving both \( \cos \theta \) and \( \sin \theta \), and noticing the power of \( \cos \theta \), we use the substitution \( u = \cos \theta \). Consequently, the derivative \( du = -\sin \theta \, d\theta \).

Step 2: Rewrite the integral

Using the substitution \( u = \cos \theta \), rewrite \( \cos^3 \theta \) as \( u^3 \). The differential \( d\theta \) from the substitution becomes \( du = -\sin \theta \cdot d\theta \), so \( \sin \theta \cdot d\theta = -du \). Thus, the integral becomes \( \int u^3 (-du) \).

Step 3: Simplify and integrate

Simplify the rewritten integral to \( -\int u^3 \, du \). The integral of \( u^3 \) with respect to \( u \) is \( \frac{u^4}{4} \), according to the power rule of integration. So the integral becomes \( -\frac{u^4}{4} + C \) where \( C \) is the constant of integration.

Step 4: Substitute back to original variables

Replace \( u \) with \( \cos \theta \) to revert to the original variable. Therefore, the integral evaluates to \( -\frac{\cos^4 \theta}{4} + C \).

Key Concepts

Substitution Method

When faced with complex integrals, the substitution method can simplify the process considerably. In this technique, you replace part of the integrand with a new variable. This makes the integral easier to solve. The key steps include:

• Selecting a substitution: Pick a part of the integrand that, when substituted, will simplify the integral.
• Finding the derivative: Determine the derivative of your substitution in terms of the original variable.
• Expressing the integral: Rewrite the integral in terms of the new variable and its derivative.

In our exercise, we identify that setting \( u = \cos \theta \) simplifies the expression due to the presence of \( \cos^3 \theta \). The derivative \( du = -\sin \theta \, d\theta \) helps rewrite the integral in terms of \( u \). This leads to a simpler integral of \( \int u^3 (-du) \), focusing on a straightforward polynomial integration in terms of \( u \). Finally, don't forget to revert the variable back to the original terms once you've integrated.

Indefinite Integrals

Indefinite integrals represent a family of functions that, when differentiated, give the original function. These integrals are expressed without specific limits, and an arbitrary constant \( C \) is always added, indicating any constant can complete the original function.

• The notation: The symbol \( \int \) represents the integral, and the expression \( f(x) \, dx \) is what we're integrating.
• The antiderivative: Finding an indefinite integral means you're looking for a function whose derivative matches \( f(x) \).
• Adding the constant \( C \): Because differentiation erases constants, indefinite integrals must include this arbitrary constant to account for any original constant lost in differentiation.

In the problem provided, we're seeking the indefinite integral of \( \cos^3 \theta \sin \theta \) which through substitution simplifies. After integration, we back-substitute to return to the variable \( \theta \), and add \( C \), resulting in \( -\frac{\cos^4 \theta}{4} + C \).

Trigonometric Functions

Trigonometric functions are fundamental in integration, often requiring specific techniques to integrate effectively. These functions include sine, cosine, tangent, and their reciprocals. Here, understanding properties of these functions can influence substitution choices.

• Common formulas: Familiarizing yourself with derivatives and integrals of basic trigonometric functions saves time.
• Identities: Use trigonometric identities to transform the integral into a simpler form if needed.
• Substitution choices: When terms like \( \cos^n \theta \) or \( \sin^n \theta \) appear, strategic substitutions can drastically simplify calculations.

In our example, the trigonometric form \( \cos^3 \theta \sin \theta \) was handled by setting \( u = \cos \theta \). This choice was influenced by knowing that \( \sin \theta \, d\theta \) allows for a clean substitution with \( du \), thus simplifying the task. Mastery of these functions can streamline tackling integrals involving trigonometric terms.