Problem 58 - Thomas Calculus

Question

Use reduction formulas to evaluate the integrals. \(\int 9 \sin ^{3} \theta \cos ^{3 / 2} \theta d \theta\)

Step-by-Step Solution

Answer

\(-\frac{18}{5} \cos^{5/2} \theta + 2 \cos^{9/2} \theta + C\).

Step 1: Split the Integrand

We start by splitting the expression in the integrand into parts that are more manageable. Rewrite \( \sin^3 \theta \cos^{3/2} \theta \) as \( (\sin \theta)(\sin^2 \theta \cos^{3/2} \theta) \).

Step 2: Use Trigonometric Identities

Use the identity \( \sin^2 \theta = 1 - \cos^2 \theta \) to simplify the expression. Replace \( \sin^2 \theta \) with \( 1 - \cos^2 \theta \), resulting in \( \int 9 \sin \theta (1 - \cos^2 \theta)\cos^{3/2} \theta \,d \theta \).

Step 3: Distribute the Expression

Distribute \( \sin \theta \) over the remaining terms: \( \int 9( \sin \theta \cos^{3/2} \theta - \sin \theta \cos^{7/2} \theta) \,d \theta \). This results in two integrals: \( 9 \int \sin \theta \cos^{3/2} \theta \,d \theta - 9 \int \sin \theta \cos^{7/2} \theta \,d \theta \).

Step 4: Use Substitution Method

For the first integral \( \int \sin \theta \cos^{3/2} \theta \,d \theta \), let \( u = \cos \theta \), then \( du = -\sin \theta \,d\theta \), or \( \sin \theta \,d\theta = -du \). The integral becomes \( -\int u^{3/2} \, du \), which evaluates to \( -\frac{2}{5} u^{5/2} + C_1 \). Substitute back \( u = \cos \theta \), giving \( -\frac{2}{5} \cos^{5/2} \theta + C_1 \).

Step 5: Solve the Second Integral with Similar Substitution

For the second integral \( \int \sin \theta \cos^{7/2} \theta \,d \theta \), use the same substitution method. Substitute \( u = \cos \theta \), giving \( -\int u^{7/2} \, du \). This evaluates to \( -\frac{2}{9} u^{9/2} + C_2 \), or \( -\frac{2}{9} \cos^{9/2} \theta + C_2 \) after substituting back.

Step 6: Combine Integrals and Simplify

Take the results from Step 4 and Step 5, and substitute these into the expression from Step 3. We get:\[9\left(-\frac{2}{5}\cos^{5/2}\theta\right) - 9\left(-\frac{2}{9}\cos^{9/2}\theta\right)\]which simplifies to:\[-\frac{18}{5} \cos^{5/2} \theta + 2 \cos^{9/2} \theta + C\]

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in calculus, particularly when simplifying complex trigonometric integrals. One common identity used in integration is \[ \sin^2 \theta = 1 - \cos^2 \theta \]. This equation simplifies expressions involving sine and cosine functions. Additionally, identities often help reduce the power of trigonometric expressions, as demonstrated in our exercise where \( \sin^2 \theta \) was replaced to facilitate integration.

• Simplifying expressions using identities like \( \sin^2 \theta = 1 - \cos^2 \theta \) allows us to transform the integrals into forms that are easier to integrate.
• Identities can convert complicated trigonometric functions into basic forms, reducing the degree of the polynomial or trigonometric power.

Understanding these identities can significantly streamline the problem-solving process in integral calculus, enhancing your ability to manage intricate functions with confidence.

Integration by Substitution

Integration by substitution is a powerful technique often used in integral calculus to make problems more tractable. This method resembles the reverse application of the chain rule from differentiation. In the exercise, we use this technique by letting \( u = \cos \theta \) and transforming the differential, \( du = -\sin \theta \, d\theta \).

• The substitution simplifies the integral from a complex expression involving products of trigonometric functions to a straightforward polynomial in terms of \( u \).
• By substituting, we reduce the power of trigonometric functions and manage the integration in a straightforward manner.
• This can often require multiple substitutions, or the inverse of the substitution, to completely solve the problem.

The goal is to transform the integral into a more familiar form, allowing for simple integration and then substituting back to the original variable for the final result.

Integral Calculus

Integral calculus is the mathematical study dealing with integrals and their properties or applications. It is a core component of calculus and facilitates solving problems related to areas, volumes, displacement, and other quantities.

• The process of integration finds functions whose derivative is the given function, known as antiderivatives or indefinite integrals.
• In definite integrals, integration calculates the numerical value representing the area under the curve of a function.
• The approach integrates by reducing complex integrals into simpler forms through techniques like substitution or using trigonometric identities.

Through systematic reduction and simplification, integral calculus helps address increasingly complex mathematical models and real-life scenarios, underlining its importance not just as a mathematical concept but as a tool for scientific inquiry.