Problem 285 - Calculus Volume 1

Question

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int\left(1-\cos ^{3} \theta\right)^{10} \cos ^{2} \theta \sin \theta d \theta $$

Step-by-Step Solution

Answer

The indefinite integral is \(-\frac{1}{33} (1 - \cos^3 \theta)^{11} + C\).

Step 1: Identify a Suitable Substitution

To simplify the integral, let's use a substitution. Notice that the expression inside the parentheses, \(1 - \cos^3 \theta\), makes a good candidate for substitution. Set \(u = 1 - \cos^3 \theta\).

Step 2: Differentiate the Substitution

Find the derivative of \(u = 1 - \cos^3 \theta\). We need \(du\) in terms of \(d\theta\). Start by differentiating \( \cos^3 \theta \) using the chain rule. We get:\[u = \cos^3 \theta \Rightarrow \frac{d}{d\theta}(u) = 3\cos^2\theta (-\sin\theta) = -3\cos^2\theta \sin\theta\].Thus, \(du = 3\cos^2\theta\sin\theta\,d\theta\).

Step 3: Solve for dθ

From \(du = -3\cos^2\theta\sin\theta\,d\theta\), solve for \(d\theta\):\[d\theta = -\frac{1}{3}\frac{du}{\cos^2\theta \sin \theta}\].We isolate \(d\theta\) in terms of \(du\) because we ultimately want it to replace the \(d\theta\) in the integral.

Step 4: Substitute Back into the Integral

Replace \((1 - \cos^3 \theta)^3\) with \(u^{10}\) and \(\cos^2 \theta \sin \theta d\theta\) with \(-\frac{1}{3} du\). Thus, the integral becomes:\[\int u^{10} \left(-\frac{1}{3}\right) du = -\frac{1}{3} \int u^{10} \, du\].

Step 5: Integrate with Respect to u

Now, integrate \(-\frac{1}{3} \int u^{10} \, du\):\[-\frac{1}{3} \left(\frac{u^{11}}{11}\right) = -\frac{1}{3} \cdot \frac{1}{11} u^{11}\]. Simplify to get \[-\frac{1}{33} u^{11}\].

Step 6: Reverse the Substitution

Recall that \(u = 1 - \cos^3 \theta\). Substitute back to express the answer in terms of \(\theta\):\[-\frac{1}{33} (1 - \cos^3 \theta)^{11} + C\], where \(C\) is the constant of integration.

Key Concepts

Substitution Method

When we tackle more complex integrals, the substitution method often comes to our rescue. It's a lot like unraveling a knot by finding one thread and pulling gently until it smooths out. Substitution involves replacing a complicated part of an integral with a simpler variable. In our exercise, we chose to set \( u = 1 - \cos^3 \theta \), which simplifies the integral significantly. This choice turns the tricky portion into something more manageable. To use substitution effectively, follow these steps:

• Identify a part of the integrand that, when replaced, simplifies the expression.
• Substitute it with a new variable, such as \( u \).
• Compute \( du \) by taking the derivative of the substitution with respect to \( \theta \).

Doing so helps us focus on the core structure of the integral, often making it closer to basic forms we're well-equipped to solve.

Chain Rule

The chain rule is a fundamental tool in calculus used to differentiate compositions of functions. It's like peeling an onion, layer by layer. To apply the chain rule, we differentiate the outer function while multiplying by the derivative of the inner function. In our example, the function \( \cos^3 \theta \) is a composition: the cube function \( (\cos \theta)^3 \). As we differentiate, we first handle the cube, then the cosine function. So, we find:\[ \frac{d}{d\theta}(\cos^3\theta) = 3\cos^2\theta (-\sin\theta) = -3\cos^2\theta \sin \theta \]The result of this application gives us \( du \) in terms of \( d\theta \), which we use to relate the original variable \( \theta \) to \( u \). This step is essential when employing substitution, as it ensures that the equation remains balanced and correct.

Trigonometric Integration

Trigonometric integration involves integrals that contain trigonometric functions, like sine and cosine. These integrals often require creative techniques, such as identities or substitutions, to solve effectively. Our specific problem involves the cosine function, showing up both cubed and squared.Here are some handy strategies for trigonometric integrals:

• Examine the expression for potential trigonometric identities that might simplify it, such as \( \sin^2 \theta + \cos^2 \theta = 1 \).
• Use substitution to convert integrated trigonometric functions into algebraic ones that are easier to integrate.

In our exercise, we handled the integral by both implementing the chain rule and using a prudent substitution that turned the trigonometric portions into a polynomial-like form. This transformation is a common and powerful tactic.

Change of Variables

Change of variables is a strategy similar to substitution, but with a slightly broader perspective. It's akin to switching viewpoints to understand a landscape better. By changing variables, we can often simplify an integral, mirroring how a new perspective brings clarity. In our problem, we changed variables from \( \theta \) to \( u \) by setting \( u = 1 - \cos^3 \theta \). This shift transforms the complicated trigonometric expression into a straightforward polynomial.Steps for effective variable change include:

• Select an inner part of the integrand to replace, such as a complex function.
• Express the differential \( d\theta \) in terms of the new variable \( du \) to facilitate integration.

This change offers fresh insight into the integral, making it more straightforward and accessible for integration using standard techniques.