Problem 83 - Thomas Calculus

Question

a. Evaluate \(\int \cos ^{3} \theta d \theta\) . (Hint: \(\cos ^{2} \theta=1-\sin ^{2} \theta . )\) b. Evaluate \(\int \cos ^{5} \theta d \theta\) c. Without actually evaluating the integral, explain how you would evaluate \(\int \cos ^{9} \theta d \theta\)

Step-by-Step Solution

Answer

a. \( \sin \theta - \frac{1}{3} \sin^3 \theta + C \); b. Use substitution and integrate term by term; c. Express using \( \cos \theta \) and substitute \( u = \sin \theta \).

Step 1: Use Trigonometric Identity

The first step for part (a) is to simplify the integral using the identity \( \cos^2 \theta = 1 - \sin^2 \theta \). First rewrite \( \cos^3 \theta \) as \( \cos \theta \cdot \cos^2 \theta \). Then replace \( \cos^2 \theta \) with \( 1 - \sin^2 \theta \): \[ \cos^3 \theta = \cos \theta (1 - \sin^2 \theta) \].

Step 2: Substitute and Simplify

To further simplify, substitute \( u = \sin \theta \), which implies \( du = \cos \theta d\theta \). The integral becomes: \[ \int \cos \theta (1 - \sin^2 \theta) \, d\theta = \int (1 - u^2) \, du \].

Step 3: Integrate

Now integrate with respect to \( u \):\[ \int (1 - u^2) \, du = \int 1 \, du - \int u^2 \, du = u - \frac{u^3}{3} + C \], where \( C \) is the constant of integration.

Step 4: Back-Substitute

Replace \( u = \sin \theta \) back into the expression to get the result in terms of \( \theta \):\[ \sin \theta - \frac{1}{3} \sin^3 \theta + C \]. This is the antiderivative of \( \cos^3 \theta \).

Step 5: Approach for \( \int \cos^5 \theta \, d\theta \)

Use similar steps as before by rewriting \( \cos^5 \theta = \cos \theta \cdot (1 - \sin^2 \theta)^2 \) and substituting \( u = \sin \theta \). Integrate \( \cos \theta (1 - \sin^2 \theta)^2 \) by expanding to get \( \int (1 - 2u^2 + u^4) \, du \), then integrate term by term. Back-substitute after integration.

Step 6: Explanation for \( \int \cos^9 \theta \, d\theta \)

To evaluate \( \int \cos^9 \theta \, d\theta \) without actually computing it, you can use the same strategy of expressing \( \cos^9 \theta \) as \( \cos \theta \cdot (1 - \sin^2 \theta)^4 \). Substitute \( u = \sin \theta \), which turns the integral into a polynomial in \( u \). This substitution simplifies the integration into manageable polynomial terms.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in calculus that allow you to simplify expressions involving trigonometric functions. These identities relate values of sine, cosine, tangent, and other trigonometric functions to each other, providing a straightforward pathway to solve integral problems.

One identity that frequently comes into play is the Pythagorean identity, such as \( \cos^2 \theta = 1 - \sin^2 \theta \). This particular identity is immensely useful when you're dealing with expressions like \( \cos^3 \theta \) because it allows you to express \( \cos^2 \theta \) in terms of \( \sin^2 \theta \).

Using trigonometric identities effectively requires practice, as recognizing when and how to apply them can convert a seemingly complex integral into something much more manageable. They help you transform the integral into a form that can be more easily integrated using algebraic and calculus techniques.

Substitution Method

The substitution method is a technique used to simplify the process of integration. It's typically used when dealing with integrals involving complex compositions of functions or products of functions that can be transformed into a single variable.

In the exercise, substitution is applied by letting \( u = \sin \theta \), making \( du = \cos \theta d\theta \). This transformation rephrases the integral into a simpler polynomial form, such as \( \int (1 - u^2) \, du \) from the original \( \int \cos^3 \theta \, d\theta \).

Substitution is particularly useful for solving integrals involving trigonometric functions, as it enables you to handle the integral in terms of \( u \), thereby opening the pathway to polynomial integration. Always remember to revert the substitution after integrating to express the solution in terms of the original variable.

Polynomial Integration

Once you’ve simplified an integral using trigonometric identities and substitution, the next step is often polynomial integration. Polynomial integration involves integrating a polynomial function term by term.

For example, when integrating the expression \( \int (1 - u^2) \, du \), you treat each term separately: \(
\begin{itemize}
\item \( \int 1 \, du \) becomes \( u \)
\item \( \int u^2 \, du \) becomes \( \frac{u^3}{3} \)
\end{itemize}\) Combining these yields \( u - \frac{u^3}{3} + C \), where \( C \) is the constant of integration.

Polynomial integration is straightforward, but don't forget the crucial step of back-substituting the original trigonometric variable, such as \( \sin \theta \), to ensure the final answer reflects the given integral's terms. This approach simplifies the integration process while maintaining accuracy throughout.